EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 14, No. 2, 2021, 551-577 ISSN 1307-5543 – ejpam.com Published by New York Business Global φ-Prime and φ-Primary Elements in Lattice Modules Ashok V. Bingi1,∗, C. S. Manjarekar2 1 Department of Mathematics, St. Xavier’s College (autonomous), Mumbai–400001, Maharashtra, India 2 Formerly at Department of Mathematics, Shivaji University, Kolhapur–416004, Maharashtra, India Abstract. In this paper, we introduce φ-prime and φ-primary elements in an L-module M . Many of its characterizations and properties are obtained. By counter examples, it is shown that a φ- prime element of M need not be prime, a φ-primary element of M need not be φ-prime, a φ- primary element of M need not be prime and a φ-primary element of M need not be primary. Finally, some results for almost prime and almost primary elements of an L-module M with their characterizations are obtained. Also, we introduce the notions of n-potent prime(respectively n-potent primary) elements in L and M to obtain interrelations among them where n > 2. 2020 Mathematics Subject Classifications: 06D10, 06E10, 06E99, 06F10, 06F99 Key Words and Phrases: φ-prime element, φ-primary element, almost prime element, almost primary element, n-potent prime element, n-potent primary element 1. Introduction In multiplicative lattices, the study of φ-prime and φ-primary elements is done by C. S. Manjarekar and A. V. Bingi in [16]. Our aim is to extend the notion of φ-prime and φ-primary elements in a multiplicative lattice to the notion of φ-prime and φ-primary elements in a lattice module and study its properties. According to [1], a proper element N of an L-module M is said to be prime if for all A ∈ M, a ∈ L, aA 6 N implies either A 6 N or a 6 (N : IM ). According to [10], a proper element N of an L-module M is said to be primary if for all A ∈M, a ∈ L, aA 6 N implies either A 6 N or a 6 √ N : IM . By restricting where aA lies, weakly prime and weakly primary elements in lattice modules are studied by C. S. Manjarekar et. al. in [19] and [20], respectively. A proper element N of an L-module M is said to be weakly prime if for all A ∈ M, a ∈ L, OM 6= aA 6 N implies either A 6 N or a 6 (N : IM ). A proper element N of an L-module M is said to be weakly primary if for all A ∈ M, a ∈ L, OM 6= aA 6 N implies either A 6 N ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v14i2.2366 Email addresses: ashok.bingi@xaviers.edu (Ashok V. Bingi), csmanjrekar@yahoo.co.in (C. S. Manjarekar) http://www.ejpam.com 551 c© 2021 EJPAM All rights reserved. A. V. Bingi, C. S. Manjarekar / Eur. J. Pure Appl. Math, 14 (2) (2021), 551-577 552 or a 6 √ N : IM . Keeping this in mind, in this paper we define and study φ-prime and φ-primary elements of an L-module M . A multiplicative lattice L is a complete lattice provided with commutative, associative and join distributive multiplication in which the largest element 1 acts as a multiplicative identity. An element e ∈ L is called meet principal if a∧ be = ((a : e)∧ b)e for all a, b ∈ L. An element e ∈ L is called join principal if (ae ∨ b) : e = (b : e) ∨ a for all a, b ∈ L. An element e ∈ L is called principal if e is both meet principal and join principal. An element a ∈ L is called compact if for X ⊆ L, a 6 ∨X implies the existence of a finite number of elements a1, a2, · · ·, an in X such that a 6 a1∨a2∨· · ·∨an. The set of compact elements of L will be denoted by L∗. If each element of L is a join of compact elements of L, then L is called a compactly generated lattice or simply a CG-lattice. L is said to be a principally generated lattice or simply a PG-lattice if each element of L is a join of principal elements of L. Throughout this paper, L denotes a compactly generated multiplicative lattice with greatest compact element 1 in which every finite product of compact elements is compact. An element a ∈ L is said to be proper if a < 1. A proper element m ∈ L is said to be maximal if for every element x ∈ L such that m < x 6 1 implies x = 1. A proper element p ∈ L is called a prime element if ab 6 p implies a 6 p or b 6 p where a, b ∈ L and is called a primary element if ab 6 p implies a 6 p or bn 6 p for some n ∈ Z+ where a, b ∈ L∗. For a, b ∈ L, (a : b) = ∨{x ∈ L | xb 6 a}. The radical of a ∈ L is denoted by √ a and is defined as ∨{x ∈ L∗ | xn 6 a, for some n ∈ Z+}. A multiplicative lattice is called as a Noether lattice if it is modular, principally generated and satisfies the ascending chain condition. A proper element a ∈ L is said to be nilpotent if an = 0 for some n ∈ Z+. According to [9], a proper element p ∈ L is said to be almost prime if for all a, b ∈ L, ab 6 p and ab p2 implies either a 6 p or b 6 p and according to [15], a proper element p ∈ L is said to be almost primary if for all a, b ∈ L, ab 6 p and ab p2 implies either a 6 p or b 6 √ p. Further study on almost prime and almost primary elements of a multiplicative lattice L is seen in [16], [5] and [4]. According to [12], a proper element q ∈ L is said to be 2-absorbing if for all a, b, c ∈ L, abc 6 q implies either ab 6 q or bc 6 q or ca 6 q. According to [18], a proper element q ∈ L is said to be 2-absorbing primary if for all a, b, c ∈ L, abc 6 q implies either ab 6 q or bc 6 √ q or ca 6 √ q. The reader is referred to [2], [3] and [9] for general background and terminology in multiplicative lattices. Let M be a complete lattice and L be a multiplicative lattice. Then M is called L- module or module over L if there is a multiplication between elements of L and M written as aB where a ∈ L and B ∈M which satisfies the following properties: 1© (∨ α aα)A = ∨ α (aα A), 2© a(∨ α Aα) = ∨ α (a Aα), 3© (ab)A = a(bA), 4© 1A = A, 5© 0A = OM , for all a, aα , b ∈ L and A,Aα ∈ M where 1 is the supremum of L and 0 is the infimum of L. We denote by OM and IM for the least element and the greatest element of M , respectively. Elements of L will generally be denoted by a, b, c, · · · and elements of M will generally be denoted by A,B,C, · · · Let M be an L-module. For N ∈M and a ∈ L , (N : a) = ∨{X ∈M | aX 6 N}. For A,B ∈M , (A : B) = ∨{x ∈ L | xB 6 A}. If (OM : IM ) = 0, then M is called a faithful L-module. M is called a torsion free L-module if for all c ∈ L, B ∈M , cB = OM implies either B = OM or c = 0. An L-module M is called a multiplication lattice module if for A. V. Bingi, C. S. Manjarekar / Eur. J. Pure Appl. Math, 14 (2) (2021), 551-577 553 every element N ∈ M there exists an element a ∈ L such that N = aIM . By proposition 3 in [10], an L-module M is a multiplication lattice module if and only if N = (N : IM )IM ∀N ∈M . An element N ∈M is called meet principal if (b ∧ (B : N))N = bN ∧B for all b ∈ L,B ∈M . An element N ∈M is called join principal if b ∨ (B : N) = ((bN ∨B) : N) for all b ∈ L,B ∈M . An element N ∈M is said to be principal if N is both meet principal and join principal. M is said to be a PG-lattice L-module if each element of M is a join of principal elements of M . An element N ∈ M is called compact if N 6 ∨ α Aα implies N 6 Aα1 ∨ Aα2 ∨ · · · ∨ Aαn for some finite subset {α1, α2, · · ·, αn}. The set of compact elements of M is denoted by M∗. If each element of M is a join of compact elements of M , then M is called a CG-lattice L-module. An element N ∈ M is said to be proper if N < IM . A proper element N ∈ M is said to be maximal if whenever there exists an element B ∈ M such that N 6 B then either N = B or B = IM . If a proper element N ∈ M is prime, then (N : IM ) ∈ L is prime. If a proper element N ∈ M is primary, then √ N : IM ∈ L is prime. A proper element N ∈ M is said to be a radical element if (N : IM ) = √ N : IM . An L-module M is said to be Noetherian, if M satisfies the ascending chain condition, is modular and is principally generated. According to [17], a proper element Q of an L-module M is said to be 2-absorbing if for all a, b ∈ L, N ∈ M , abN 6 Q implies either ab 6 (Q : IM ) or bN 6 Q or aN 6 Q. According to [6], a proper element Q of an L-module M is said to be 2-absorbing primary if for all a, b ∈ L, N ∈M , abN 6 Q implies either ab 6 (Q : IM ) or bN 6 ( √ Q : IM )IM or aN 6 ( √ Q : IM )IM . The reader is referred to [1], [10] and [14] for terminology in lattice modules. This paper is motivated by [24] and [7]. Many of the results obtained in this paper are lattice module version of the results in [16] and principal elements of M are used wherever needed with some more conditions on M . First section of this paper is comprised of φ- prime and φ-primary elements of an L-module M . Second section is comprised of almost prime and almost primary elements of an L-module M . By counter examples, it is shown that a φ-prime element of M need not be prime (see Example 1), a φ-primary element of M need not be φ-prime (see Example 2), a φ-primary element of M need not be prime (see Example 3) and a φ-primary element of M need not be primary (see Example 4). We define 2-potent prime and 2-potent primary elements in an L-module M . By counter examples, it is shown that an almost primary element of M need not be 2-potent prime (see Example 5) and a 2-potent prime element of M which is almost primary need not be prime (see Example 6). Also, we introduce the notions of n-potent prime and n-potent primary elements in an L-module M where n > 2. We find condition(s) under which a φ-prime element of M is prime (see Theorems 5-10). Also, we find condition(s) under which a φ-primary element of M is primary (see Theorems 15-23). Absorbing concepts in an L-module M are related to these notions of φ-prime and φ-primary in M . In the last section of this paper, many characterizations of almost prime and almost primary elements of M are obtained. By a counter example, it is shown that an almost primary element of M need not be idempotent (see Example 7). By a counter example, it is shown that an almost primary element of M need not be weakly primary (see Example 8). Finally, we show that if an element in M is almost prime (respectively almost primary), then its corresponding element in L is also almost prime (respectively almost primary) and vice A. V. Bingi, C. S. Manjarekar / Eur. J. Pure Appl. Math, 14 (2) (2021), 551-577 554 versa. 2. φ-Prime and φ-Primary Elements in M The study of weakly prime and weakly primary elements of an L-module M is carried out by A. V. Bingi and C. S. Manjarekar in [8]. Also, the notion of an almost prime element of an L-module M is seen in [22]. With weakly prime elements and almost prime elements of an L-module M in mind, we begin with introducing the notion of a φ-prime element of an L-module M . Definition 1. Let φ : M −→ M be a function on an L-module M . A proper element N ∈ M is said to be φ-prime if for all a ∈ L, A ∈ M , aA 6 N and aA φ(N) implies either A 6 N or a 6 (N : IM ). Now if φα : M −→M is a function on an L-module M , then φα-prime elements of M are defined by following settings in the Definition 1 of a φ-prime element. • φ0(N) = OM . Then N ∈M is called a weakly prime element. • φ2(N) = (N : IM )N . Then N ∈M is called a 2-almost prime element or a φ2-prime element or simply an almost prime element. • φn(N) = (N : IM )n−1N (n > 2). Then N ∈M is called an n-almost prime element or a φn-prime element (n > 2). • φω(N) = ∧∞ i=1(N : IM )iN . Then N ∈ M is called a ω-prime element or φω-prime element. Since N\φ(N) = N\(N ∧ φ(N)), so without loss of generality, throughout this paper, we assume that φ(N) 6 N . Definition 2. Given two functions γ1, γ2 : M −→ M on an L-module M , we define γ1 6 γ2 if γ1(N) 6 γ2(N) for all N ∈M . Clearly, we have the following order: φ0 6 φω 6 · · · 6 φn+1 6 φn 6 · · · 6 φ2 Now before obtaining the characterizations of a φ-prime element of an L-module M , we state the following essential lemma which is outcome of Lemma 2.3.13 from [11]. Lemma 1. Let a1, a2 ∈ L. Suppose b ∈ L satisfies the following property: (∗). If h ∈ L∗ with h 6 b, then either h 6 a1 or h 6 a2. Then either b 6 a1 or b 6 a2. Theorem 1. Let M be a CG-lattice L-module, N ∈M be a proper element and φ : M −→ M be a function on M . Then the following statements are equivalent: 1© N is a φ-prime element of M . A. V. Bingi, C. S. Manjarekar / Eur. J. Pure Appl. Math, 14 (2) (2021), 551-577 555 2© For every A ∈M such that A N , either (N : A) = (N : IM ) or (N : A) = (φ(N) : A). 3© For every r ∈ L such that r (N : IM ), either (N : r) = N or (N : r) = (φ(N) : r). 4© For every r ∈ L∗, A ∈M∗, if rA 6 N and rA φ(N), then either r 6 (N : IM ) or A 6 N . Proof. 1©=⇒ 2©. Suppose 1© holds. Let A ∈ M be such that A N . Obviously, (φ(N) : A) 6 (N : A) and (N : IM ) 6 (N : A). Let a ∈ L∗ be such that a 6 (N : A). Then aA 6 N . If aA 6 φ(N), then a 6 (φ(N) : A). If aA φ(N), then since N is φ-prime and A N , it follows that a 6 (N : IM ). Hence by Lemma 1, either (N : A) 6 (φ(N) : A) or (N : A) 6 (N : IM ). Thus either (N : A) = (φ(N) : A) or (N : A) = (N : IM ). 2©=⇒ 3©. Suppose 2© holds. Let r (N : IM ) for r ∈ L. Then rIM N . Using 2©, we have, either (N : rIM ) = (N : IM ) or (N : rIM ) = (φ(N) : rIM ). Now let K 6 (N : r) for K ∈ M∗. As (K : IM )IM 6 K, we have, (K : IM )IM 6 (N : r) and (K : IM )IM ∈ M∗. Clearly, K 6 (N : r) implies (K : IM ) 6 ((N : r) : IM ) = (N : rIM ). So we have either (K : IM ) 6 (N : IM ) or (K : IM ) 6 (φ(N) : rIM ) = (φ(N) : r : IM ). This gives either (K : IM )IM 6 N or (K : IM )IM 6 (φ(N) : r). This implies that either (N : r) 6 N or (N : r) 6 (φ(N) : r), by Lemma 3.1 of [22]. Since rN 6 N gives N 6 (N : r) and φ(N) 6 N gives (φ(N) : r) 6 (N : r), it follows that either (N : r) = N or (N : r) = (φ(N) : r). 3©=⇒ 4©. Suppose 3© holds. Let rA 6 N , rA φ(N) and r (N : IM ) for r ∈ L∗, A ∈ M∗. Then by 3©, we have either (N : r) = (φ(N) : r) or (N : r) = N . If (N : r) = (φ(N) : r), then as rA 6 N , it follows that A 6 (φ(N) : r) which contradicts rA φ(N) and so we must have (N : r) = N . Therefore rA 6 N gives A 6 N . 4©=⇒ 1©. Suppose 4© holds. Let aQ 6 N , aQ φ(N) and Q N for a ∈ L, Q ∈M . As L and M are compactly generated, there exist x′ ∈ L∗ and Y, Y ′ ∈ M∗ such that x′ 6 a, Y 6 Q, Y ′ 6 Q, Y ′ N and x′Y ′ φ(N). Let x ∈ L∗ be such that x 6 a. Then (x∨x′) ∈ L∗, (Y ∨Y ′) ∈M∗ such that (x∨x′)(Y ∨Y ′) 6 aQ 6 N, (x∨x′)(Y ∨Y ′) φ(N) and (Y ∨ Y ′) N . So by 4©, (x ∨ x′) 6 (N : IM ) which implies a 6 (N : IM ). Therefore N is φ-prime. The following 2 corollaries are consequences of Theorem 1. Corollary 1. Let M be a CG-lattice L-module and N ∈ M be a proper element. Then the following statements are equivalent: 1© N is a weakly prime element of M . 2© For every A ∈ M such that A N , either (N : A) = (N : IM ) or (N : A) = (OM : A). 3© For every r ∈ L such that r (N : IM ), either (N : r) = N or (N : r) = (OM : r). 4© For every r ∈ L∗, A ∈M∗, if OM 6= rA 6 N , then either r 6 (N : IM ) or A 6 N . A. V. Bingi, C. S. Manjarekar / Eur. J. Pure Appl. Math, 14 (2) (2021), 551-577 556 Corollary 2. Let M be a CG-lattice L-module and N ∈ M be a proper element. Then the following statements are equivalent: 1© N is an almost prime element of M . 2© For every A ∈M such that A N , either (N : A) = ((N : IM )N : A) or (N : A) = (N : IM ). 3© For every r ∈ L such that r (N : IM ), either (N : r) = ((N : IM )N : r) or (N : r) = N . 4© For every r ∈ L∗, A ∈M∗, if rA 6 N and rA (N : IM )N , then either A 6 N or r 6 (N : IM ). To obtain the relation among prime, weakly prime, ω-prime, n-almost prime (n > 2) and almost prime elements of an L-module M , we prove the following result. Theorem 2. Let γ1, γ2 : M −→ M be functions on an L-module M such that γ1 6 γ2. Then every proper γ1-prime element of M is γ2-prime. Proof. Let a proper element N ∈ M be γ1-prime. Assume that aA 6 N and aA γ2(N) for a ∈ L, A ∈M . Then as γ1 6 γ2, we have aA γ1(N). Since N is γ1-prime, it follows that either A 6 N or a 6 (N : IM ) and hence N is γ2-prime. Theorem 3. Let N be a proper element of an L-module M . Then N is prime implies N is weakly prime, N is weakly prime implies N is ω-prime, N is ω-prime implies N is n-almost prime (n > 2) and N is n-almost prime (n > 2) implies N is almost prime. Proof. By definition, every prime element of an L-module M is weakly prime and hence N is prime implies N is weakly prime. The remaining implications follow by using Theorem 2 to the fact that φ0 6 φω 6 · · · 6 φn+1 6 φn 6 · · · 6 φ2. From the Theorem 3, we get the following characterization of a ω-prime element of an L-module M . Corollary 3. Let N be a proper element of an L-module M . Then N is ω-prime if and only if N is n-almost prime for every n > 2. Proof. Assume that N ∈ M is n-almost prime for every n > 2. Let aA 6 N and aA ∧∞ i=1(N : IM )iN for a ∈ L, A ∈ M . Then aA (N : IM )n−1N for some n > 2. Since N is n-almost prime, we have either a 6 (N : IM ) or A 6 N and hence N is ω-prime. The converse follows from Theorem 3. Before going to the characterization of an n-almost prime element of an L-module M , we recall the definition of the Jacobson radical of L. According to [2], in a multiplicative lattice L with 1 compact, the Jacobson radical is the element ∧{m ∈ L | m is a maximal element}. A. V. Bingi, C. S. Manjarekar / Eur. J. Pure Appl. Math, 14 (2) (2021), 551-577 557 Theorem 4. Let L be a Noether lattice, M be a torsion free Noetherian L-module and f ∈ L be the Jacobson radical. Then a proper element N ∈ M such that (N : IM ) 6 f is n-almost prime for every n > 2 if and only if N is prime. Proof. Assume that N ∈ M is n-almost prime where n > 2. Let aA 6 N for a ∈ L, A ∈ M . If aA (N : IM )n−1N for n > 2, then as N is n-almost prime, we have either A 6 N or a 6 (N : IM ). If aA 6 (N : IM )n−1N for all n > 2, then as (N : IM ) 6 f , from Corollary 3.3 of [13], it follows that aA 6 ∧∞ n=1(N : IM )nN = OM and thus aA = OM . Since M is torsion free, we have either A = OM or a = 0 which implies either A 6 N or a 6 (N : IM ) and hence N is prime. The converse follows from Theorem 3. Clearly, every prime element of an L-module M is φ-prime. But the converse is not true which is shown in the following example by taking φ(N) = (N : IM )N for convenience. Example 1. If Z is the ring of integers, then Z24 is a Z−module. Assume that (k) denotes the cyclic ideal of Z generated by k ∈ Z and < t > denotes the cyclic submodule of Z−module Z24 where t ∈ Z24. Suppose that L = L(Z) is the set of all ideals of Z and M = L(Z24) is the set of all submodules of Z−module Z24. The multiplication between elements of L and M is given by (ki) < tj >=< kitj > for every (ki) ∈ L and < tj >∈M where ki, tj ∈ Z. Then M is a lattice module over L [[22], Example 2.5]. Let N be the cyclic submodule of M generated by 0. It is easy to see that OM =< 0 >= N is weakly prime and hence almost prime (φ2-prime) while N is not prime, since (2) < 12 >6 N but < 12 > N and (2) (N : IM ) = (0) where IM =< 1 >. Now we obtain six results that show under which condition(s) a φ-prime element of an L-module M is prime. But before that we prove the required cancellation laws of M in the form of following lemmas. Lemma 2. Let M be a torsion free L-module and OM 6= A ∈M be a weak join principal element. Then aA 6 bA implies a 6 b for a, b ∈ L where b 6= 0. Proof. Let aA 6 bA and OM 6= A ∈M be a weak join principal element for a, b ∈ L. As M is a torsion free L-module, we have (OM : A) = 0. Then clearly, a = a ∨ 0 = a ∨ (OM : A) = (aA : A) 6 (bA : A) = b ∨ (OM : A) = b ∨ 0 = b which implies a 6 b. Lemma 3. Let M be a torsion free L-module and OM 6= A ∈M be a weak join principal element. Then aA = bA implies a = b for a, b ∈ L where a 6= 0, b 6= 0. Proof. The proof is obvious. Now we have a characterization of a φ-prime element of an L-module M . Theorem 5. Let M be a torsion free L-module and OM 6= N < IM be a weak join principal element of M . Then N is φ-prime for some φ 6 φ2 if and only if N is prime. A. V. Bingi, C. S. Manjarekar / Eur. J. Pure Appl. Math, 14 (2) (2021), 551-577 558 Proof. Assume that N ∈ M is a prime element. Then obviously, N is φ-prime for every φ and hence for some φ 6 φ2. Conversely, let N be φ-prime for some φ 6 φ2. Then by Theorem 2, N is φ2-prime. Let aA 6 N for a ∈ L, A ∈M . If aA φ2(N), then as N is φ2-prime, we have either A 6 N or a 6 (N : IM ). Next, assume that aA 6 φ2(N). If a(A∨N) φ2(N), then as a(A∨N) 6 N and N is φ2-prime, we have either (A∨N) 6 N or a 6 (N : IM ) and hence either A 6 N or a 6 (N : IM ). Finally, if a(A ∨N) 6 φ2(N), then aN 6 (N : IM )N which implies a 6 (N : IM ), by Lemma 2 and hence N is prime. Now we show that the Theorem 5 can also be achieved by changing the conditions on M and L. According to [23], in a Noether lattice L, an element a ∈ L is said to satisfy the restricted cancellation law (RCL) if for all b, c ∈ L, ab = ac 6= 0 implies b = c. Theorem 6. Let L be a Noether PG-lattice and M be a faithful multiplication PG-lattice L-module with IM compact. Let N be a proper element of M such that 0 6= (N : IM ) ∈ L satisfies the restricted cancellation law (RCL) and is a non-nilpotent element. Then N is φ-prime for some φ 6 φ2 if and only if N is prime. Proof. Assume that N ∈ M is a prime element. Then obviously, N is φ-prime for every φ and hence for some φ 6 φ2. Conversely, let N be φ-prime for some φ 6 φ2. Then by Theorem 2, N is φ2-prime. Let aA 6 N for a ∈ L, A ∈M . If aA φ2(N), then as N is φ2-prime, we have either A 6 N or a 6 (N : IM ). Next, assume that aA 6 φ2(N). If a(A∨N) φ2(N), then as a(A∨N) 6 N and N is φ2-prime, we have either (A∨N) 6 N or a 6 (N : IM ) and hence either A 6 N or a 6 (N : IM ). Finally, if a(A∨N) 6 φ2(N), then aN 6 (N : IM )N which implies a(N : IM )IM 6 (N : IM )2IM , since M is a multiplication lattice L module. As IM is compact, this gives a(N : IM ) 6 (N : IM )2 6= 0, by Theorem 5 of [10]. This implies a 6 (N : IM ), by Lemma 1.11 of [23] and hence N is prime. Now we define a 2-potent prime element in an L-module M . Definition 3. A proper element N ∈M is said to be 2-potent prime if for all a ∈ L, A ∈ M , aA 6 (N : IM )N implies either a 6 (N : IM ) or A 6 N . Theorem 7. Let a proper element N of an L-module M be 2-potent prime. Then N is φ-prime for some φ 6 φ2 if and only if N is prime. Proof. Assume that N ∈ M is a prime element. Then obviously, N is φ-prime for every φ and hence for some φ 6 φ2. Conversely, let N be φ-prime for some φ 6 φ2. Then by Theorem 2, N ∈ M is φ2-prime. Let aA 6 N for a ∈ L, A ∈ M . If aA (N : IM )N , then as N is φ2-prime, we have either a 6 (N : IM ) or A 6 N . If aA 6 (N : IM )N , then as N is 2-potent prime, we have either a 6 (N : IM ) or A 6 N and hence N is prime. Now we define a n-potent prime element in an L-module M where n > 2. Definition 4. Let n > 2 and n ∈ Z+. A proper element N ∈ M is said to be n-potent prime if for all a ∈ L, A ∈ M , aA 6 (N : IM )n−1N implies either a 6 (N : IM ) or A 6 N . A. V. Bingi, C. S. Manjarekar / Eur. J. Pure Appl. Math, 14 (2) (2021), 551-577 559 Theorem 8. A proper element N of an L-module M is φ-prime for some φ 6 φn where n > 2 if and only if N is prime, provided N is k-potent prime for some k 6 n. Proof. Assume that N ∈ M is a prime element. Then obviously, N is φ-prime for every φ and hence for some φ 6 φn where n > 2. Conversely, let N be φ-prime for some φ 6 φn where n > 2. Then by Theorem 2, N ∈ M is φn-prime. Let aA 6 N for a ∈ L, A ∈M . If aA φk(N), then aA φn(N) as k 6 n. Since N is φn-prime, we have either a 6 (N : IM ) or A 6 N . If aA 6 φk(N), then as N is k-potent prime, we have either a 6 (N : IM ) or A 6 N and hence N is prime. The following corollary is outcome of Theorems 5, 6 and 7. Corollary 4. An almost prime element N of an L-module M is prime if one the following statements hold true: (i) M is torsion free and OM 6= N < IM is a weak join principal element. (ii) N is a 2-potent prime element. (iii) L is a Noether PG-lattice, M is a faithful multiplication PG-lattice with IM compact, 0 6= (N : IM ) ∈ L satisfies the restricted cancellation law (RCL) and is a non- nilpotent element. Theorem 9. Let a proper element N of an L-module M be φ-prime. If φ(N) is prime, then N is prime. Proof. Let aA 6 N for a ∈ L, A ∈ M . If aA φ(N), then as N is φ-prime, we have either a 6 (N : IM ) or A 6 N and we are done. If aA 6 φ(N), then as φ(N) is prime, we have either aIM 6 φ(N) or A 6 φ(N). This implies that either aIM 6 N or A 6 N because φ(N) 6 N . Hence N is prime. Theorem 10. Let a proper element N of an L-module M be φ-prime. If (N : IM )N φ(N), then N is prime. Proof. Let aA 6 N for a ∈ L, A ∈ M . If aA φ(N), then as N is φ-prime, we have either a 6 (N : IM ) or A 6 N . So assume that aA 6 φ(N). First suppose aN φ(N). Then aN0 φ(N) for some N0 6 N in M . Since N is φ-prime, a(A∨N0) = aA∨aN0 6 N and a(A ∨ N0) φ(N), we have either a 6 (N : IM ) or (A ∨ N0) 6 N and hence either a 6 (N : IM ) or A 6 N . Next, assume that aN 6 φ(N). If (N : IM )A φ(N), then k0A φ(N) for some k0 6 (N : IM ) in L. Since N is φ-prime, (a ∨ k0)A 6 N and (a∨k0)A φ(N), we have either (a∨k0) 6 (N : IM ) or A 6 N and hence either a 6 (N : IM ) or A 6 N . Now let (N : IM )A 6 φ(N). By hypothesis, as (N : IM )N φ(N), there exist k 6 (N : IM ) in L and N0 6 N in M such that kN0 φ(N). Since N is φ-prime, (a ∨ k)(A ∨N0) 6 N and (a ∨ k)(A ∨N0) φ(N), we have either (a ∨ k) 6 (N : IM ) or (A ∨N0) 6 N and hence either a 6 (N : IM ) or A 6 N . Therefore N is prime. The consequences of Theorem 10 are presented in the following corollaries. A. V. Bingi, C. S. Manjarekar / Eur. J. Pure Appl. Math, 14 (2) (2021), 551-577 560 Corollary 5. If a proper element N of a multiplication lattice L-module M is φ-prime but not prime, then (N : IM )2IM 6 φ(N). Proof. Since M is a multiplication lattice L-module, by Proposition 3 of [10], we have N = (N : IM )IM . So (N : IM )2IM = (N : IM )N 6 φ(N) by Theorem 10. Corollary 6. If a proper element N of an L-module M is weakly prime such that (N : IM )N 6= OM , then N is prime. Proof. The proof is obvious. Corollary 7. If a proper element N of an L-module M is φ-prime such that φ 6 φ3, then N is ω-prime. Proof. If N is prime, then by Theorem 3, N is ω-prime. So assume that N is not prime. Then by Theorem 10 and hypothesis, we get (N : IM )2N 6 (N : IM )N 6 φ(N) 6 (N : IM )2N and so φ(N) = (N : IM )2N = (N : IM )N . Now consider (N : IM )3N = ((N : IM )(N : IM )2)N = (N : IM )((N : IM )2N) = (N : IM )((N : IM )N) = ((N : IM )(N : IM ))N = (N : IM )2N = φ(N) and so on. Hence φ(N) = (N : IM )n−1N for every n > 2. Consequently, N is n-almost prime for every n > 2 and thus N is ω-prime by Corollary 3. Corollary 8. If a proper element N of a multiplication lattice L-module M is φ-prime but not prime, then √ N : IM = √ φ(N) : IM . Proof. By Corollary 5, we have (N : IM )2IM 6 φ(N) which implies (N : IM ) 6√ φ(N) : IM . Hence √ N : IM 6 √√ φ(N) : IM = √ φ(N) : IM , by property (p3) of radicals in [21]. Also, as φ(N) 6 N , we have √ φ(N) : IM 6 √ N : IM and thus √ N : IM =√ φ(N) : IM . Corollary 9. If a proper element N of a multiplication lattice L-module M is φ-prime, then either √ φ(N) : IM 6 (N : IM ) or (N : IM ) 6 √ φ(N) : IM . Proof. The proof is obvious. Now we introduce the notion of φ-primary element of an L-module M . Definition 5. Let φ : M −→ M be a function on an L-module M . A proper element N ∈M is said to be φ-primary if for all a ∈ L, A ∈M , aA 6 N and aA φ(N) implies either A 6 N or an 6 (N : IM ) for some n ∈ Z+. Now if φα : M −→ M is a function on an L-module M , then φα-primary elements of M are defined by following settings in the Definition 5 of a φ-primary element. • φ0(N) = OM . Then N ∈M is called a weakly primary element. • φ2(N) = (N : IM )N . Then N ∈ M is called a 2-almost primary element or a φ2-primary element or simply an almost primary element. A. V. Bingi, C. S. Manjarekar / Eur. J. Pure Appl. Math, 14 (2) (2021), 551-577 561 • φn(N) = (N : IM )n−1N (n > 2). Then N ∈ M is called an n-almost primary element or a φn-primary element (n > 2). • φω(N) = ∧∞ i=1(N : IM )iN . Then N ∈ M is called a ω-primary element or φω- primary element. Clearly, every φ-prime element of an L-module M is φ-primary but the converse is not true as shown in the following example by taking φ(N) = (N : IM )N for convenience. Example 2. Consider the lattice module as in Example 1. Let N be the cyclic submodule of M generated by 4. It is easy to see that the element N =< 4 > is almost primary (φ2-primary) but N is not almost prime (φ2-prime) because (2) < 6 >6 N , (2) < 6 > φ2(N) =< 8 > but < 6 > N and (2) (N : IM ) = (4) where IM =< 1 >. Clearly, every prime element of an L-module M is φ-primary. But the converse is not true which is shown in the following example by taking φ(N) = (N : IM )N for convenience. Example 3. Consider the lattice module as in Example 1. Let N be the cyclic submodule of M generated by 0. It is easy to see that the element N =< 0 >= OM is almost primary (φ2-primary) but N is not prime. The analogous results (from the results of φ-prime elements of M) for φ-primary ele- ments of M are stated below whose proofs being on similar arguments are omitted. We begin with the characterizations of a φ-primary element of an L-module M . Theorem 11. Let M be a CG-lattice L-module, N ∈ M be a proper element and φ : M −→M be a function on M . Then the following statements are equivalent: (i) N is a φ-primary element of M . (ii) For every A ∈M such that A N , either (N : A) 6 √ N : IM or (N : A) = (φ(N) : A). (iii) For every r ∈ L such that r √ N : IM , either (N : r) = N or (N : r) = (φ(N) : r). (iv) For every r ∈ L∗, A ∈M∗, if rA 6 N and rA φ(N), then either r 6 √ N : IM or A 6 N . The following 2 corollaries are consequences of Theorem 11. Corollary 10. Let M be a CG-lattice L-module and N ∈ M be a proper element. Then the following statements are equivalent: 1© N is a weakly primary element of M . 2© For every A ∈ M such that A N , either (N : A) 6 √ N : IM or (N : A) = (OM : A). 3© For every r ∈ L such that r √ N : IM , either (N : r) = N or (N : r) = (OM : r). A. V. Bingi, C. S. Manjarekar / Eur. J. Pure Appl. Math, 14 (2) (2021), 551-577 562 4© For every r ∈ L∗, A ∈M∗, if OM 6= rA 6 N , then either r 6 √ N : IM or A 6 N . Corollary 11. Let M be a CG-lattice L-module and N ∈ M be a proper element. Then the following statements are equivalent: 1© N is an almost primary element of M . 2© For every A ∈M such that A N , either (N : A) = ((N : IM )N : A) or (N : A) 6√ N : IM . 3© For every r ∈ L such that r √ N : IM , either (N : r) = ((N : IM )N : r) or (N : r) = N . 4© For every r ∈ L∗, A ∈ M∗, if rA 6 N and rA (N : IM )N , then either r 6√ N : IM or A 6 N . To obtain the relation among primary, weakly primary, ω-primary, n-almost primary (n > 2) and almost primary elements of an L-module M , we have the following result. Theorem 12. Let γ1, γ2 : M −→M be functions on an L-module M such that γ1 6 γ2. Then every proper γ1-primary element of M is γ2-primary. Theorem 13. Let N be a proper element of an L-module M . Then N is primary implies N is weakly primary, N is weakly primary implies N is ω-primary, N is ω-primary implies N is n-almost primary (n > 2), N is n-almost primary (n > 2) implies N is almost primary. From the Theorem 13, we get the following characterization of a ω-primary element of an L-module M . Corollary 12. Let N ∈M be a proper element of an L-module M . Then N is ω-primary if and only if N is n-almost primary for every n > 2. The following theorem gives the characterization of an n-almost primary element of an L-module M . Theorem 14. Let L be a Noether lattice, M be a torsion free Noetherian L-module and f ∈ L be the Jacobson radical. Then a proper element N ∈ M such that (N : IM ) 6 f is n-almost primary for every n > 2 if and only if N is primary. Clearly, every primary element of an L-module M is φ-primary. But the converse is not true which is shown in the following example by taking φ(N) = (N : IM )N for convenience. Example 4. If Z is the ring of integers, then Z30 is a Z−module. Assume that (k) denotes the cyclic ideal of Z generated by k ∈ Z and < t > denotes the cyclic submodule of Z−module Z30 where t ∈ Z30. Suppose that L = L(Z) is the set of all ideals of Z and M = L(Z30) is the set of all submodules of Z−module Z30. The multiplication between A. V. Bingi, C. S. Manjarekar / Eur. J. Pure Appl. Math, 14 (2) (2021), 551-577 563 elements of L and M is given by (ki) < tj >=< kitj > for every (ki) ∈ L and < tj >∈M where ki, tj ∈ Z. Then M is a lattice module over L. Let N be the cyclic submodule of M generated by 6. It is easy to see that N =< 6 > is almost primary (φ2-primary) while N is not primary, since (3) < 2 >6 N but < 2 > N and (3)n (N : IM ) = (6) for every n ∈ Z+ where IM =< 1 >. In the following successive nine theorems, we show under which condition(s) a φ- primary element of an L-module M is primary. Now we have a characterization of a φ-primary element of an L-module M . Theorem 15. Let M be a torsion free L-module and OM 6= N < IM be a weak join principal element of an L-module M . Then N is φ-primary for some φ 6 φ2 if and only if N is primary. The following result shows that the Theorem 15 can also be achieved by changing the conditions on M and L. Theorem 16. Let L be a Noether PG-lattice and M be a faithful multiplication PG-lattice L-module with IM compact. Let N be a proper element of M such that 0 6= (N : IM ) ∈ L satisfies the restricted cancellation law (RCL) and is a non-nilpotent element. Then N is φ-primary for some φ 6 φ2 if and only if N is primary. Now we define a 2-potent primary element in an L-module M . Definition 6. A proper element N ∈ M is said to be 2-potent primary if for all a ∈ L, A ∈M , aA 6 (N : IM )N implies either A 6 N or am 6 (N : IM ) for some m ∈ Z+. Theorem 17. Let a proper element N of an L-module M be 2-potent primary. Then N is φ-primary for some φ 6 φ2 if and only if N is primary. Clearly, every 2-potent prime element of an L-module M is 2-potent primary. Theorem 18. Let a proper element N of an L-module M be 2-potent prime. Then N is φ-primary for some φ 6 φ2 if and only if N is primary. Now we define a n-potent primary element in an L-module M where n > 2. Definition 7. Let n > 2 and n ∈ Z+. A proper element N ∈ M is said to be n- potent primary if for all a ∈ L, A ∈ M , aA 6 (N : IM )n−1N implies either A 6 N or am 6 (N : IM ) for some m ∈ Z+. Theorem 19. A proper element N of an L-module M is φ-primary for some φ 6 φn where n > 2 if and only if N is primary, provided N is k-potent primary for some k 6 n. Clearly, every n-potent prime element of an L-module M is n-potent primary. Theorem 20. A proper element N of an L-module M is φ-primary for some φ 6 φn where n > 2 if and only if N is primary, provided N is k-potent prime for some k 6 n. A. V. Bingi, C. S. Manjarekar / Eur. J. Pure Appl. Math, 14 (2) (2021), 551-577 564 The following corollary is outcome of Theorems 15, 16, 17 and 18. Corollary 13. An almost primary element N of an L-module M is primary if one the following statements hold true: (i) M is torsion free and OM 6= N < IM is a weak join principal element. (ii) N is a 2-potent primary element. (iii) N is a 2-potent prime element. (iv) L is a Noether PG-lattice, M is a faithful multiplication PG-lattice with IM compact, 0 6= (N : IM ) ∈ L satisfies the restricted cancellation law (RCL) and is a non- nilpotent element. From the following examples, it is clear that, an almost primary element of an L module M need not be 2-potent prime and a 2-potent prime element of an L module M which is almost primary need not be prime. Example 5. Consider the lattice module as in Example 4. Let N be the cyclic submodule of M generated by 6. It is easy to see that the element N =< 6 > is almost primary but not 2-potent prime. Example 6. If Z is the ring of integers, then Z8 is a Z−module. Assume that (k) denotes the cyclic ideal of Z generated by k ∈ Z and < t > denotes the cyclic submodule of Z−module Z8 where t ∈ Z8. Suppose that L = L(Z) is the set of all ideals of Z and M = L(Z8) is the set of all submodules of Z−module Z8. The multiplication between elements of L and M is given by (ki) < tj >=< kitj > for every (ki) ∈ L and < tj >∈M where ki, tj ∈ Z. Then M is a lattice module over L. Let N be the cyclic submodule of M generated by 4. It is easy to see that N =< 4 > is almost primary (φ2-primary) and 2-potent prime but not prime. Theorem 21. Let a proper element N of an L-module M be φ-primary. If φ(N) is primary, then N is primary. Theorem 22. Let a proper element N of an L-module M be φ-primary. If (N : IM )N φ(N), then N is primary. The consequences of Theorem 22 are presented in the form of following corollaries. Corollary 14. If a proper element N of a multiplication lattice L-module M is φ-primary but not primary, then (N : IM )2IM 6 φ(N). Corollary 15. If a proper element N of an L-module M is weakly primary such that (N : IM )N 6= OM , then N is primary. Corollary 16. If a proper element N of an L-module M is φ-primary such that φ 6 φ3, then N is ω-primary. A. V. Bingi, C. S. Manjarekar / Eur. J. Pure Appl. Math, 14 (2) (2021), 551-577 565 Corollary 17. If a proper element N of a multiplication lattice L-module M is φ-primary but not primary, then √ N : IM = √ φ(N) : IM . Corollary 18. If a proper element N of a multiplication lattice L-module M is φ-primary, then either √ φ(N) : IM 6 (N : IM ) or (N : IM ) 6 √ φ(N) : IM . Theorem 23. Let a proper element N of an L-module M be φ-primary. If ( √ N : IM )N φ(N), then N is primary. Proof. Just mimic the proof of Theorem 10. Now, the interrelations among prime, primary, 2-absorbing and 2-absorbing primary elements of an L-module M are given in following theorems whose proofs being obvious are omitted. Theorem 24. Every prime element of an L-module M is primary and 2-absorbing. Theorem 25. If Q is a primary element of an L-module M , then √ Q : IM is a prime element and hence a 2-absorbing element of L. Also, it is a 2-absorbing primary element of L. Theorem 26. If Q is a 2-absorbing element of an L-module M , then both √ Q : IM and (Q : IM ) are 2-absorbing elements of L. Also, they are 2-absorbing primary elements of L. Theorem 27. Let L be a PG-lattice and M be a faithful multiplication PG-lattice L- module with IM compact. If Q is a 2-absorbing primary element of M , then (Q : IM ) is a 2-absorbing primary element of L and √ Q : IM is a 2-absorbing element of L. Proof. Let abc 6 (Q : IM ) for a, b, c ∈ L. Then as ab(cIM ) 6 Q and Q is a 2-absorbing primary element of M , we have, either ab 6 (Q : IM ) or a(cIM ) 6 ( √ Q : IM )IM or b(cIM ) 6 ( √ Q : IM )IM . Since IM is compact, by Theorem 5 of [10], it follows that, either ab 6 (Q : IM ) or ac 6 √ Q : IM or bc 6 √ Q : IM and hence (Q : IM ) is a 2-absorbing primary element of L. By Theorem 2.4 in [18], it follows that √ Q : IM is a 2-absorbing element of L. By relating the absorbing concepts with φ-prime and φ-primary elements of an L- module M , we obtain the following results. Theorem 28. Let a proper element N of an L-module M be φ-prime. If (N : IM )N φ(N), then N is primary and 2-absorbing. Also, then both √ N : IM and (N : IM ) are 2-absorbing and hence 2-absorbing primary elements of L. Proof. The proof follows from Theorems 10, 24 and 26. Clearly, every primary element of a multiplication L-module M is 2-absorbing primary. Theorem 29. Let a proper element N of a multiplication L-module M be φ-prime. If (N : IM )N φ(N), then N is 2-absorbing primary. Also, then (N : IM ) is a 2-absorbing primary element of L provided M is a faithful PG-lattice with IM compact and L as a PG-lattice. Further, √ N : IM is a 2-absorbing element of L. A. V. Bingi, C. S. Manjarekar / Eur. J. Pure Appl. Math, 14 (2) (2021), 551-577 566 Proof. The proof follows from Theorems 10, 24 and 27. Theorem 30. Let a proper element N of a multiplication L-module M be φ-primary. If (N : IM )N φ(N), then N is 2-absorbing primary. Proof. The proof follows from Theorem 22. Theorem 31. Let L be a PG-lattice and M be a faithful multiplication PG-lattice L- module with IM compact. Let a proper element N of an L-module M be φ-primary. If (N : IM )N φ(N), then (N : IM ) is a 2-absorbing primary element of L and √ N : IM is a 2-absorbing element of L. Proof. The proof follows from Theorems 30 and 27. The following results are obtained by relating the absorbing concepts with almost prime and almost primary elements of an L-module M . Theorem 32. Let M be a torsion free L-module and OM 6= N < IM be a weak join principal element of M . If N is almost prime, then N is primary and 2-absorbing. Also, then both √ N : IM and (N : IM ) are 2-absorbing and hence 2-absorbing primary elements of L. Proof. The proof follows from Theorems 5, 24 and 26. Theorem 33. Let M be a torsion free, multiplication L-module and OM 6= N < IM be a weak join principal element of M . If N is almost prime, then N is 2-absorbing primary. Also, then (N : IM ) is a 2-absorbing primary element of L provided M is a faithful PG- lattice with IM compact and L as a PG-lattice. Further, √ N : IM is a 2-absorbing element of L. Proof. The proof follows from Theorems 5, 24 and 27. Theorem 34. Let M be a torsion free, multiplication L-module and OM 6= N < IM be a weak join principal element of M . If N is almost primary, then N is 2-absorbing primary. Proof. The proof follows from Theorem 15. Theorem 35. Let M be a torsion free, faithful, multiplication PG-lattice L-module with IM compact and L be a PG-lattice. Let OM 6= N < IM be a weak join principal element of M . If N is almost primary, then (N : IM ) is a 2-absorbing primary element of L and√ N : IM is a 2-absorbing element of L. Proof. The proof follows from Theorems 34 and 27. A. V. Bingi, C. S. Manjarekar / Eur. J. Pure Appl. Math, 14 (2) (2021), 551-577 567 3. Almost Prime and Almost Primary Elements in M In this section, we will obtain some more results on an almost prime (respectively al- most primary) element of an L-module M by relating it with an idempotent element and a weakly prime (respectively weakly primary) element of an L-module M . Also, many characterizations of an almost prime and almost primary element of an L-module M are obtained. Finally, we define n-potent prime(respectively n-potent primary) elements in L and these notions are related with n-potent prime(respectively n-potent primary) elements in M where n > 2. Clearly, every almost prime element of an L-module M is almost primary but the converse need not be true as seen in Example 2. It is easy to see that converse holds for radical elements of an L-module M . Every prime element of an L-module M is almost prime and every primary element of an L-module M is almost primary but their converses are not true as seen in Example 1 and Example 4, respectively. Also, every prime element of an L-module M is almost primary. According to Definition 2.6 of [22], an idempotent element of an L-module M is defined in the following way. Definition 8. A proper element N of an L-module M is said to be idempotent if (N : IM )N = N . Clearly, every idempotent element of an L-module M is almost prime and hence almost primary. But an almost primary element of an L-module M need not be idempotent as shown in the following example. Example 7. Consider the lattice module as in Example 6. Let N be the cyclic submodule of M generated by 4. It is easy to see that the element N =< 4 > is almost primary but not idempotent. Theorem 36. Let L be a PG-lattice and M be a faithful multiplication PG-lattice L- module with IM compact. For an idempotent element N ∈ M , ( √ (N : IM )N : IM )N = (N : IM )N . Proof. As N < IM is idempotent, N is almost prime (φ2 − prime). Since M is a multiplication lattice L-module, we have (N : IM )2IM = (N : IM )N which implies (N : IM ) 6 √ (N : IM )N : IM . Thus (N : IM )N 6 ( √ (N : IM )N : IM )N . Now to prove that ( √ (N : IM )N : IM )N 6 (N : IM )N , let a 6 √ (N : IM )N : IM for a ∈ L. If a 6 (N : IM ), then we are done. So let a (N : IM ). Then as N is φ2 − prime, by Theorem 1, we have either (N : a) = N or (N : a) = ((N : IM )N : a). Let (N : a) = N and n be the least positive integer such that an 6 ((N : IM )N : IM ). If n = 1, then aIM 6 (N : IM )N = (N : IM )2IM . As IM is compact, by Theorem 5 of [10], we have a 6 (N : IM )2 6 (N : IM ) which contradicts a (N : IM ). So assume that n > 2. Then anIM 6 (N : IM )N 6 N with akIM (N : IM )N for every k 6 (n − 1). Since A. V. Bingi, C. S. Manjarekar / Eur. J. Pure Appl. Math, 14 (2) (2021), 551-577 568 a(an−1IM ) 6 N , we have an−1IM 6 (N : a) = N with an−1IM (N : IM )N . If n = 2, then aIM 6 N which contradicts a (N : IM ). If n > 3, then a(an−2IM ) 6 N but a(an−2IM ) (N : IM )N . As N is almost prime, we have either a 6 (N : IM ) or an−2IM 6 N . As a 6 (N : IM ) is a contradiction, let an−2IM 6 N . Then a(an−3IM ) 6 N but a(an−3IM ) (N : IM )N . As N is almost prime, we have either a 6 (N : IM ) or an−3IM 6 N . Continuing this process we conclude that a 6 (N : IM ) which contradicts a (N : IM ). Hence we must have (N : a) = ((N : IM )N : a). Then aN 6 a(N : a) = a((N : IM )N : a) 6 (N : IM )N which implies a 6 ((N : IM )N : N) and so√ (N : IM )N : IM 6 ((N : IM )N : N). It follows that ( √ (N : IM )N : IM )N 6 (N : IM )N and hence ( √ (N : IM )N : IM )N = (N : IM )N . From following example, it is clear that an almost primary element of an L-module M need not be weakly primary. Example 8. Consider the lattice module as in Example 4. Let N be the cyclic submodule of M generated by 6. It is easy to see that the element N =< 6 > is almost primary (φ2-primary) but not weakly primary. Before obtaining the characterization of an almost primary element of an L-module M in terms of a weakly primary element of M , we recall the definition of a local module M . According to [1], an L-module M is said to be a local module if it has a unique maximal element. Theorem 37. Let M be a local L-module with a unique maximal element Q ∈ M such that (Q : IM )Q = OM . Then a proper element N ∈M is almost primary if and only if N is weakly primary. Proof. Assume that a proper element N ∈ M is almost primary. Then N 6 Q. It follows that (N : IM )N 6 (Q : IM )Q = OM and hence (N : IM )N = OM . Let OM 6= aA 6 N for a ∈ L, A ∈ M . As aA 6 N , aA (N : IM )N = OM and N is almost primary, we have either A 6 N or a 6 √ N : IM and hence N is weakly primary. The converse is obvious from Theorem 13. Now we prove the result required to show that if an element in M (or L) is almost primary, then its corresponding element in L (or M) is also almost primary. Lemma 4. Let M be a torsion free multiplication lattice L-module and IM be a weak join principal element of M . Let N be a proper element of M . Then a(N : IM ) = (aN : IM ) for a ∈ L. Proof. Since M is a multiplication lattice L-module, N = (N : IM )IM . Then a(N : IM )IM = aN = (aN : IM )IM and so the result follows by Lemma 3. Theorem 38. Let L be a PG-lattice and M be a faithful multiplication torsion free PG- lattice L-module with IM compact. Let IM be a weak join principal element and N be a proper element of M . Then the following statements are equivalent: A. V. Bingi, C. S. Manjarekar / Eur. J. Pure Appl. Math, 14 (2) (2021), 551-577 569 1© N is an almost primary element of M . 2© (N : IM ) is an almost primary element of L. 3© N = qIM for some almost primary element q ∈ L which is maximal in the sense that if aIM = N , then a 6 q where a ∈ L. Proof. 1©=⇒ 2©. Assume that N is an almost primary element of M . Let ab 6 (N : IM ) and ab (N : IM )2 for a, b ∈ L. Then abIM 6 N . If abIM 6 (N : IM )N , then by Lemma 4, we have ab 6 ((N : IM )N : IM ) = (N : IM )(N : IM ) which contradicts ab (N : IM )2. So let a(bIM ) (N : IM )N . Then as N is almost primary, we have either a 6 √ N : IM or bIM 6 N and thus (N : IM ) is an almost primary element of L. 2©=⇒ 3©. Assume that (N : IM ) = q is an almost primary element of L. Then qIM 6 N . Since M is a multiplication lattice module, N = aIM for some a ∈ L. So a 6 (N : IM ) = q and thus N = aIM 6 qIM . Hence N = qIM for some almost primary element q ∈ L which is maximal in the sense that if aIM = N , then a 6 q. 3©=⇒ 1©. Suppose N = qIM for some almost primary element q ∈ L which is maximal in the sense that if aIM = N , then a 6 q where a ∈ L. Then q 6 (N : IM ). Now, let rX 6 N , rX (N : IM )N and X N for r ∈ L, X ∈ M . Since M is a multiplication lattice module, X = cIM for some c ∈ L. Then rc 6 (N : IM ) 6 q, using maximality of q to N = (N : IM )IM (by Proposition 3 of [10]). If rc 6 q2, then rX 6 qN 6 (N : IM )N , a contradiction. So rc q2. Also, c q because if c 6 q, then X 6 N , a contradiction. Now, as rc 6 q, rc q2, c q and q is almost primary, we have, r 6 √ q which implies r 6 √ N : IM and hence N is almost primary Theorem 39. Let L be a PG-lattice and M be a faithful multiplication torsion free PG- lattice L-module with IM compact. Let IM be a weak join principal element and N be a proper element in M . Then the following statements are equivalent: 1© N is an almost primary element of M . 2© (N : IM ) is an almost primary element of L. 3© N = qIM for some almost primary element q ∈ L. Proof. 1©=⇒ 2© follows from 1©=⇒ 2© in the proof of Theorem 38. 2©=⇒ 1©. Assume that (N : IM ) is an almost primary element of L. Let rQ 6 N and rQ (N : IM )N for r ∈ L, Q ∈M . Then (rQ : IM ) 6 (N : IM ) and so by Lemma 4, we have r(Q : IM ) = (rQ : IM ) 6 (N : IM ). If r(Q : IM ) 6 (N : IM )2 = ((N : IM )N : IM ), then r(Q : IM )IM 6 (N : IM )N which implies rQ 6 (N : IM )N , a contradiction. If r(Q : IM ) (N : IM )2, then as r(Q : IM ) 6 (N : IM ) and (N : IM ) is almost primary, we have either r 6 √ N : IM or (Q : IM ) 6 (N : IM ) which implies either r 6 √ N : IM or Q 6 N and thus N is an almost primary element of M . 2©=⇒ 3©. Suppose (N : IM ) is an almost primary element of L. Since M is a multipli- cation lattice L-module, N = (N : IM )IM and hence 3© holds. A. V. Bingi, C. S. Manjarekar / Eur. J. Pure Appl. Math, 14 (2) (2021), 551-577 570 3©=⇒ 2©. Suppose N = qIM for some almost primary element q ∈ L. As M is a multiplication lattice L-module, N = (N : IM )IM . Since IM is compact, 2© holds by Theorem 5 of [10]. Now we relate the almost primary element N ∈ M with rad(N) ∈ M , the radical of N . According to definition 3.1 in [17], the radical of a proper element N in an L module M is defined as ∧{P ∈M | P is a prime element and N 6 P} and is denoted as rad(N). Using Theorem 3.6 of [17], we have the following interesting characterization of an almost primary element of M . Theorem 40. Let L be a PG-lattice and M be a faithful multiplication torsion free PG- lattice L-module with IM compact. Let IM ∈M be a weak join principal element. Then a proper element P ∈M is almost primary (φ2−primary) if and only if whenever N = aIM and K = bIM in M are such that abIM 6 P and abIM (P : IM )P then either N 6 P or K 6 rad(P ) for a, b ∈ L. Proof. Assume that P ∈ M is almost primary. Let N = aIM and K = bIM in M be such that abIM 6 P and abIM (P : IM )P for a, b ∈ L. Since M is a multiplication lattice L-module, we have a = (N : IM ) and b = (K : IM ) and so (K : IM )(N : IM )IM = abIM 6 P and (K : IM )(N : IM )IM (P : IM )P . As P ∈ M is almost primary, we have either (N : IM )IM 6 P or (K : IM ) 6 √ P : IM which implies either N = (N : IM )IM 6 P or K = (K : IM )IM 6 ( √ P : IM )IM = rad(P ) by Theorem 3.6 of [17]. Conversely, assume that abIM 6 P and abIM (P : IM )P implies either N 6 P or K 6 rad(P ) where N = aIM and K = bIM are in M for a, b ∈ L. Let rs 6 (P : IM ) and rs (P : IM )2 where S = rIM and Q = sIM are in M for r, s ∈ L. If rsIM 6 (P : IM )P , then since M is a multiplication lattice L-module, we have rsIM 6 (P : IM )2IM . So by Theorem 5 of [10], we have rs 6 (P : IM )2, a contradiction. So let rsIM (P : IM )P . Since rsIM 6 P , by hypothesis, we have either S 6 P or Q 6 rad(P ) which implies either rIM 6 P or sIM 6 rad(P ) = ( √ P : IM )IM , by Theorem 3.6 of [17]. So either r 6 (P : IM ) or s 6 √ P : IM , by Theorem 5 of [10]. Thus (P : IM ) is an almost primary element of L and hence by Theorem 39, P is an almost primary element of M . Now we show that Lemma 4 can also be achieved by changing the conditions on M and IM . Lemma 5. Let L be a PG-lattice and M be a faithful multiplication PG-lattice L-module with IM compact. Let N be a proper element of M . Then a(N : IM ) = (aN : IM ) for a ∈ L. Proof. Since M is a multiplication lattice L-module, N = (N : IM )IM . Then a(N : IM )IM = aN = (aN : IM )IM and we are done, by Theorem 5 of [10]. Lemma 5 is Lemma 3.5 of [22]. In view of Lemma 5, the Theorems 38, 39 and 40 can be restated in the following way. A. V. Bingi, C. S. Manjarekar / Eur. J. Pure Appl. Math, 14 (2) (2021), 551-577 571 Theorem 41. Let L be a PG-lattice and M be a faithful multiplication PG-lattice L- module with IM compact. Let N be a proper element of an L-module M . Then the following statements are equivalent: 1© N is an almost primary element of M . 2© (N : IM ) is an almost primary element of L. 3© N = qIM for some almost primary element q ∈ L which is maximal in the sense that if aIM = N , then a 6 q where a ∈ L. Theorem 42. Let L be a PG-lattice and M be a faithful multiplication PG-lattice L- module with IM compact. Let N be a proper element of an L-module M . Then the following statements are equivalent: 1© N is an almost primary element of M . 2© (N : IM ) is an almost primary element of L. 3© N = qIM for some almost primary element q ∈ L. Theorem 43. Let L be a PG-lattice and M be a faithful multiplication PG-lattice L- module with IM compact. Then a proper element P ∈M is almost primary (φ2−primary) if and only if whenever N = aIM and K = bIM in M are such that abIM 6 P and abIM (P : IM )P then either N 6 P or K 6 rad(P ) for a, b ∈ L. The following result is a consequence of the Theorem 42. Corollary 19. Let L be a PG-lattice and M be a faithful multiplication PG-lattice L- module with IM compact. Then a proper element N of an L-module M is almost primary if and only if (N : IM ) is an almost primary element of L. The analogous results (from the results of almost primary elements of M) for almost prime elements of M are as follows. In Example 2.5 of [22], it is shown that an almost prime element of an L-module M need not be weakly prime. The following characterization of an almost prime element of an L-module M shows that under a certain condition, an almost prime element of an L-module M is weakly prime. Theorem 44. Let M be a local L-module with a unique maximal element Q ∈ M such that (Q : IM )Q = OM . Then a proper element N ∈M is almost prime if and only if N is weakly prime. Proof. Assume that a proper element N ∈M is almost prime. Then N 6 Q. It follows that (N : IM )N 6 (Q : IM )Q = OM and hence (N : IM )N = OM . Let OM 6= aA 6 N for a ∈ L, A ∈ M . As aA 6 N , aA (N : IM )N = OM and N is almost prime, we have either A 6 N or a 6 (N : IM ) and hence N is weakly prime. The converse is obvious from Theorem 3. The following result shows that if an element in M (or L) is almost prime, then its corresponding element in L (or M) is also almost prime. A. V. Bingi, C. S. Manjarekar / Eur. J. Pure Appl. Math, 14 (2) (2021), 551-577 572 Theorem 45. Let L be a PG-lattice and M be a faithful multiplication torsion free PG- lattice L-module with IM compact. Let IM be a weak join principal element and N be a proper element of M . Then the following statements are equivalent: 1© N is an almost prime element of M . 2© (N : IM ) is an almost prime element of L. 3© N = qIM for some almost prime element q ∈ L which is maximal in the sense that if aIM = N , then a 6 q where a ∈ L. Proof. 1©=⇒ 2©. Assume that N is an almost prime element of M . Let ab 6 (N : IM ) and ab (N : IM )2 for a, b ∈ L. Then abIM 6 N . If abIM 6 (N : IM )N , then by Lemma 4, we have ab 6 ((N : IM )N : IM ) = (N : IM )(N : IM ) which contradicts ab (N : IM )2. So let a(bIM ) (N : IM )N . Then as N is almost prime, we have either a 6 (N : IM ) or bIM 6 N and thus (N : IM ) is an almost prime element of L. 2©=⇒ 3©. Assume that (N : IM ) = q is an almost prime element of L. Then qIM 6 N . Since M is a multiplication lattice module, N = aIM for some a ∈ L. So a 6 (N : IM ) = q and thus N = aIM 6 qIM . Hence N = qIM for some almost prime element q ∈ L which is maximal in the sense that if aIM = N , then a 6 q. 3©=⇒ 1©. Suppose N = qIM for some almost prime element q ∈ L which is maximal in the sense that if aIM = N , then a 6 q where a ∈ L. Then q 6 (N : IM ). Now, let rX 6 N , rX (N : IM )N and X N for r ∈ L, X ∈ M . Since M is a multiplication lattice module, X = cIM for some c ∈ L. Then rc 6 (N : IM ) 6 q, using maximality of q to N = (N : IM )IM (by Proposition 3 of [10]). If rc 6 q2, then rX 6 qN 6 (N : IM )N , a contradiction. So rc q2. Also, c q because if c 6 q, then X 6 N , a contradiction. Now, as rc 6 q, rc q2, c q and q is almost prime, we have, r 6 q which implies r 6 (N : IM ) and hence N is almost prime Theorem 46. Let L be a PG-lattice and M be a faithful multiplication torsion free PG- lattice L-module with IM compact. Let IM be a weak join principal element and N be a proper element of M . Then the following statements are equivalent: 1© N is an almost prime element of M . 2© (N : IM ) is an almost prime element of L. 3© N = qIM for some almost prime element q ∈ L. Proof. 1©=⇒ 2© follows from 1©=⇒ 2© in the proof of Theorem 45. 2©=⇒ 1©. Assume that (N : IM ) is an almost prime element of L. Let rQ 6 N and rQ (N : IM )N for r ∈ L, Q ∈M . Then (rQ : IM ) 6 (N : IM ) and so by Lemma 4, we have r(Q : IM ) = (rQ : IM ) 6 (N : IM ). If r(Q : IM ) 6 (N : IM )2 = ((N : IM )N : IM ), then r(Q : IM )IM 6 (N : IM )N which implies rQ 6 (N : IM )N , a contradiction. If r(Q : IM ) (N : IM )2, then as r(Q : IM ) 6 (N : IM ) and (N : IM ) is almost prime, we A. V. Bingi, C. S. Manjarekar / Eur. J. Pure Appl. Math, 14 (2) (2021), 551-577 573 have either r 6 (N : IM ) or (Q : IM ) 6 (N : IM ) which implies either r 6 (N : IM ) or Q 6 N and thus N is an almost prime element of M . 2©=⇒ 3©. Suppose (N : IM ) is an almost prime element of L. Since M is a multiplica- tion lattice L-module, N = (N : IM )IM and hence 3© holds. 3©=⇒ 2©. Suppose N = qIM for some almost prime element q ∈ L. As M is a multiplication lattice L-module, N = (N : IM )IM . Since IM is compact, 2© holds by Theorem 5 of [10]. The following result is another characterization of an almost prime element of an L- module M . Theorem 47. Let L be a PG-lattice and M be a faithful multiplication torsion free PG- lattice L-module with IM compact. Let IM be a weak join principal element. Then a proper element P ∈ M is almost prime (φ2 − prime) if and only if whenever N = aIM and K = bIM in M are such that abIM 6 P and abIM (P : IM )P then either N 6 P or K 6 P for a, b ∈ L. Proof. Assume that P ∈M is almost prime. Let N = aIM and K = bIM in M be such that abIM 6 P and abIM (P : IM )P for a, b ∈ L. Since M is a multiplication lattice L- module, we have a = (N : IM ) and b = (K : IM ) and so (K : IM )(N : IM )IM = abIM 6 P and (K : IM )(N : IM )IM (P : IM )P . As P ∈ M is almost prime, we have either (N : IM )IM 6 P or (K : IM ) 6 (P : IM ) which implies either N = (N : IM )IM 6 P or K = (K : IM )IM 6 P . Conversely, assume that abIM 6 P and abIM (P : IM )P implies either N 6 P or K 6 P where N = aIM and K = bIM are in M for a, b ∈ L. Let rs 6 (P : IM ) and rs (P : IM )2 where S = rIM and Q = sIM are in M for r, s ∈ L. If rsIM 6 (P : IM )P , then since M is a multiplication lattice L-module, we have rsIM 6 (P : IM )2IM . So by Theorem 5 of [10], we have rs 6 (P : IM )2, a contradiction. So let rsIM (P : IM )P . Since rsIM 6 P , by hypothesis, we have either S 6 P or Q 6 P which implies either rIM 6 P or sIM 6 P and so either r 6 (P : IM ) or s 6 (P : IM ). Thus (P : IM ) is an almost prime element of L and hence by Theorem 46, P is an almost prime element of M . In view of Lemma 5, the Theorems 45, 46 and 47 can be restated in the following way. Theorem 48. Let L be a PG-lattice and M be a faithful multiplication PG-lattice L- module with IM compact. Let N be a proper element of an L-module M . Then the following statements are equivalent: 1© N is an almost prime element of M . 2© (N : IM ) is an almost prime element of L. 3© N = qIM for some almost prime element q ∈ L which is maximal in the sense that if aIM = N , then a 6 q where a ∈ L. Theorem 49. Let L be a PG-lattice and M be a faithful multiplication PG-lattice L- module with IM compact. Let N be a proper element of an L-module M . Then the following statements are equivalent: A. V. Bingi, C. S. Manjarekar / Eur. J. Pure Appl. Math, 14 (2) (2021), 551-577 574 1© N is an almost prime element of M . 2© (N : IM ) is an almost prime element of L. 3© N = qIM for some almost prime element q ∈ L. Theorem 49 is Theorem 3.8 of [22]. Theorem 50. Let L be a PG-lattice and M be a faithful multiplication PG-lattice L- module with IM compact. Then a proper element P ∈ M is almost prime (φ2 − prime) if and only if whenever N = aIM and K = bIM in M are such that abIM 6 P and abIM (P : IM )P then either N 6 P or K 6 P for a, b ∈ L. Theorem 50 is Theorem 3.14 of [22]. The following result is a consequence of the Theorem 49. Corollary 20. Let L be a PG-lattice and M be a faithful multiplication PG-lattice L- module with IM compact. Then a proper element N of an L-module M is almost prime if and only if (N : IM ) is an almost prime element of L. According to [16], a proper element q ∈ L is said to be 2-potent prime if for all a, b ∈ L, ab 6 q2 implies either a 6 q or b 6 q and a proper element q ∈ L is said to be 2-potent primary if for all a, b ∈ L, ab 6 q2 implies either a 6 q or b 6 √ q. In view of these definitions, we define n-potent prime and n-potent primary elements (where n > 2) in a multiplicative lattice L in following way. Definition 9. Let n > 2 and n ∈ Z+. A proper element q ∈ L is said to be n-potent prime if for all a, b ∈ L, ab 6 qn implies either a 6 q or b 6 q. Definition 10. Let n > 2 and n ∈ Z+. A proper element q ∈ L is said to be n-potent primary if for all a, b ∈ L, ab 6 qn implies either a 6 q or b 6 √ q. Now we show that if an element in M is n-potent prime (respectively n-potent pri- mary), then its corresponding element in L is also n-potent prime (respectively n-potent primary) and vice-versa where n > 2. Theorem 51. Let L be a PG-lattice and M be a faithful multiplication PG-lattice L- module with IM compact. Let N be a proper element of an L-module M and n > 2. Then the following statements are equivalent: 1© N is a n-potent prime element of M . 2© (N : IM ) is a n-potent prime element of L. 3© N = qIM for some n-potent prime element q ∈ L. A. V. Bingi, C. S. Manjarekar / Eur. J. Pure Appl. Math, 14 (2) (2021), 551-577 575 Proof. Since M is a multiplication lattice L-module, by Proposition 3 of [10], we have N = (N : IM )IM . 1©=⇒ 2©. Assume that N is a n-potent prime element of M . Let ab 6 (N : IM )n for a, b ∈ L. Then a(bIM ) 6 (N : IM )n−1N . As N is n-potent prime, we have either a 6 (N : IM ) or bIM 6 N and thus (N : IM ) is a n-potent prime element of L. 2©=⇒ 1©. Assume that (N : IM ) is a n-potent prime element of L. Let aX 6 (N : IM )n−1N for a ∈ L and X ∈M . M being a multiplication lattice L-module, we have X = cIM for some c ∈ L. Clearly, a(cIM ) 6 (N : IM )nIM . This implies that ac 6 (N : IM )n by Theorem 5 of [10]. As (N : IM ) is a n-potent prime, we have either a 6 (N : IM ) or c 6 (N : IM ) which implies either a 6 (N : IM ) or X = cIM 6 (N : IM )IM = N and thus N is a n-potent prime element of M . 2©=⇒ 3©. Suppose q = (N : IM ) is a n-potent prime element of L. Since M is a multiplication lattice L-module, N = (N : IM )IM = qIM and hence 3© holds. 3©=⇒ 2©. Suppose N = qIM for some n-potent prime element q ∈ L. As M is a multiplication lattice L-module, N = (N : IM )IM . Since IM is compact, 2© holds by Theorem 5 of [10]. Theorem 52. Let L be a PG-lattice and M be a faithful multiplication PG-lattice L- module with IM compact. Let N be a proper element of an L-module M and n > 2. Then the following statements are equivalent: 1© N is a n-potent primary element of M . 2© (N : IM ) is a n-potent primary element of L. 3© N = qIM for some n-potent primary element q ∈ L. Proof. Just mimic the proof of Theorem 51. We conclude this paper with following 2 results which are outcomes of Theorems 51 and 52, respectively. Corollary 21. Let L be a PG-lattice and M be a faithful multiplication PG-lattice L- module with IM compact. Then a proper element N of an L-module M is 2-potent prime if and only if (N : IM ) is a 2-potent prime element of L. Corollary 22. Let L be a PG-lattice and M be a faithful multiplication PG-lattice L- module with IM compact. Then a proper element N of an L-module M is 2-potent primary if and only if (N : IM ) is a 2-potent primary element of L. 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