Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol. 32 No. 2s (2024) 538 https://internationalpubls.com Pure and Weakly Pure Elements in Lattice Modules Santosh Mitkari1, Renu Pathak2, Smita Nigam3, Pradip Girase4, Lakpa Sherpa5, Narayan Phadatare6 1, 2Department of Mathematics, School of Science, Sandip University, Nashik (INDIA) 3Genba Sopanrao Moze College of Engineering, Pune (INDIA) 4Department of Mathematics, K. K. M. College Manwath, Parbhani (INDIA) 5Department of Mathematics, Savitribai Phule Pune University, Pune-411 007 (INDIA) 6Bharati Vidyapeeth Deemed to be University College of Engineering, Pune-411 043 (INDIA) santosh.mitkari@bharatividyapeeth.edu, renu.pathak@sandipuniversity.edu.in, smita.nigam03@gmail.com, gpradipmaths22@gmail.com, csherpaap@gmail.com, nmphadatare@bvucoep.edu.in Article History: Received: 25-09-2024 Revised: 02-11-2024 Accepted: 13-11-2024 Abstract: This study concerns with investigation of Pure and Weakly pure elements of lattice modules. An element N of M is called pure, if aN = N ∧ a1M , for each a of L. An element K of M is called weakly pure, if aN = N ∧ a1M , for each idempotent element a of L. Also, this study obtains the relation between pure, idempotent and multiplication elements of lattice modules. Keywords: Pure element, Weakly Pure element, Idempotent element, Multiplication element. 1. Introduction A lattice L is called as a multiplicative lattice, if L is complete with commutative, associative and join distributive binary operation called as multiplication. An element 1L of L act as a identity with respect to multiplication. For a1, a2 ∈ L, (a1 : a2) = ∨{x ∈ L|a2x ≤ a1}. Element p ∈ L such that p ≠1L is prime, if p1.p2 ≤ p implies p1 ≤ p or p2 ≤ p. The radical of a ∈ Lis denoted by √𝑎 and is defined as ∨{x ∈ L|xk ≤ a, for some k ∈ Z+} = ∧{p ∈ L|a ≤ p and p is a prime element}. An element c ∈ L is called compact, if for t ∈ I(I is an index set), c ≤ ∨tat ⇒ c ≤ ⋁𝑖=0 𝑛 𝑎𝑡𝑖 , for some n ∈ Z+. If each element of L is a join of compact elements of L, then L is called a CG-lattice. An element p ∈ L is called meet [join] principal, if a1 ∧ a2p=((a1 : p) ∧ a2)p [((a1p ∨ a2) : p) = a1 ∨ (a2 : p)], ∀ a1, a2 ∈ L. If p ∈ L is both meet and join principal, then p is called principal element. If every element of L is a join of principal elements of L, then L is called a PG-lattice. An element p ∈ L is said to be weak meet [join] principal, if a ∧ p = p(a : p) [a ∨ (0L : p) = (pa : p)], ∀ a ∈ L. An element a ∈ L is called semiprime or radical, if √𝑎 = a. If a ∈ L such that a2 = a, then a is called an idempotent. Let c ∈ L. If for each a ∈ L such that a ≤ c there exists an element d ∈ L such that a = cd, then c is called multiplication element. Note that, a ∈ L is a multiplication element if and only if it is weak meet principal element in L. A complete lattice M is called a lattice module (L-module), where L is a multiplicative lattice, if the multiplication aN ∈ M , for a ∈ L and N ∈ M satisfies,(ab)N = a(bN ); for all a, Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol. 32 No. 2s (2024) 539 https://internationalpubls.com b in L and for all N in M . 1. (∨α lα)( ∨β Nβ) = (∨αβ lαNβ); for all lα in L and for all Nβ in M . 2. 1LN = N ; for 1L ∈ L and N ∈ M . 3. 0LN = 0M ; for 0L ∈ L and N ∈ M . Note that 0M is a least and 1M is a greatest element of M. For N1,N2 ∈ M , (N1 : N2) = ∨{x ∈ L|xN2 ≤ N1}. For N ∈ M and a ∈ L, (N: a) = ∨{K ∈ M |aK ≤ N }. An element N∈ M is called compact, if for t ∈ I(I is an index set), N ≤ ∨tBt ⇒ N≤ ∨𝑖=0 𝑛 𝐵𝑡𝑖 , for some n ∈ Z+. If each element of M is a join of compact elements of M, then M is called a CG- lattice module. An element N ∈ M is called meet [join] principal, if (a ∧ (B : N ))N =aN ∧ B [(a ∨ (B : N )=((aN ∨ B) : N )], ∀ a ∈ L and B ∈ M. If B ∈ M is both meet and join principal, then B is called principal element. If each element of M is a join of principal elements of M, then M is called a PG-lattice module. An element N ∈ M is said to be weak meet [join] principal, if (B : N )N = B ∧ N [(aN : N ) = a ∨ (0M : N )], ∀ a ∈ L and B ∈ M . An element N ∈ M is said to be proper, if N < 1M. If N ∈ M such that N = (N : 1M )N, then N is an idempotent element of M . Element N ∈ M is said to be multiplication, if for every K ∈ M with K ≤ N there exists an element a ∈ L such that K = aN. It is also noted that, N ∈ M is a multiplication element if and only if N is weak meet principal in M . A L−lattice module M is called second, if for each a ∈ L, a1M = 1M or a1M = 0M. A L−lattice module M is called secondary, if for each a ∈ L, a1M = 1M or an1M = 0M for some n>0. If annM = (0M : 1M ) = 0L, then M is called faithful L−module. A L−module M is called torsion-free, whenever aK = 0M implies K = 0M or a = 0L, for any a ∈ L and K ∈ M. A L−module M is multiplication, if for each element N ∈ M there exists a ∈ L such that N = a1M. Note that, L−module M is a multiplication if and only if N = (N : 1M )1M for all N ∈ M (see [4]). For N ∈ M, [N, 1M ] is a set of all K ∈ M such that N ≤ K ≤ 1M. Note that, [N, 1M ] is a L-lattice module with multiplication a ◦ K = aK ∨ N, where a ∈ L and K ∈ M such that N ≤ K. This study aims the generalization of some important results studied in [1], [2] for submodules of module over commutative ring to the lattice modules over multiplicative lattices and examine the concepts in multiplicative lattices and multiplication lattice modules. Remark 1.1. Let M be a multiplication lattice module and N a element of M. If (N : 1M ) is an idempotent, then N = (N : 1M )1M = (N : 1M )21M = (N : 1M )N , and N is idempotent in M . Conversely, if M is a CG and faithful multiplication L-module with N is idempotent in M, then N = (N : 1M )1M = (N : 1M )N, and hence N = (N : 1M )21M = (N : 1M )1M, which shows that (N : 1M )2 = (N : 1M ) is an idempotent. Further, for more information on modules, multiplicatice lattices, lattice modules, the reader may refer to [3], [7], [8], [9], [10]. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol. 32 No. 2s (2024) 540 https://internationalpubls.com 2. Pure Element We begin this section with the following definitions: Definition 2.1. Let L be a multiplicative lattice and c ∈ L. c is said to be a multiplication element, if for every element a of L such that a ≤ c there exists an element d ∈ L such that a = cd. [5]. Definition 2.2. [5] Let L be a multiplicative lattice and M a lattice L-module. N ∈ M is said to be a multiplication element, if for every element K of M such that K ≤ N there exists an element a ∈ L such that K = aN. Definition 2.3. Let L be a multiplicative lattice and M a lattice L-module. N ∈ M is said to be a idempotent element in M, if N = (N : 1M )N. Proposition 2.4. Let L be a CG-lattice, M be a nonzero L-lattice module and 0M ≠ N is pure element of M. If M is p-secondary lattice module, then [N, 1M] and [0M, N] are both p-secondary lattice modules. Proof. see [6], Proposition 13. Proposition 2.5. Let L be a domain. If M is a multiplication second L-module, then every element in M is pure. Proof. Let N be any element of M. Since M is a multiplication second L-module, so M is either divisible or torsion [6]. If M is divisible, then a1M = 1M , for every 0L ≠ a ∈ L. So aN = N = N ∧ a1M, since M is multiplication L-module. If M is torsion, then a1M = 0M, for every 0L ≠ a ∈ L. So, aN = 0M = N ∧ a1M. Lemma 2.6. Let M be a multiplication L−module, and 0M ≠ N be a pure element of M. Then M is a p-second lattice module if and only if [0M , N ] and [N, 1M ] are both p-second lattice modules. Proof. see [6], Proposition 14. Lemma 2.7. Let M be a faithful multiplication L−module. If N is a pure element of M, then N is multiplication and is idempotent in M. Proof. Let K be a element of M. Then K = (K : 1M )1M . Since N is a pure element of M, we have, (K : N )N = N ∧ (K : N )1M ≥ N ∧ (K : 1M )1M = N ∧ K ≥ (K : N )N, so that (K : N )N = K ∧ N ⇒ N a weak meet principal element in M, and N is multiplication. Since N is pure in M, we have that (N : 1M )N = N ∧ (N : 1M )1M = N, and hence N is idempotent in M . Lemma 2.8. Let M be a multiplication L−module. If N is a pure element of M, then K = (N : 1M )K and (K : N )N = (K : 1M )N, for each K of M. Proof. By Lemma 2.7, N is multiplication and is idempotent in M. Let K ≤ N, then K = (K : N )N = (K : N )(N : 1M )N = (N : 1M )K. Also, for K ≤ N , (K : N )N = (K : N )(N : 1M )N ≤ (K : 1M )N ≤ (K : N )N , so that (K : N )N = (K : 1M )N . Lemma 2.9. Let M be a faithful multiplication L−module. If N is a pure element of M, then a(N : 1M ) = a ∧ (N : 1M ), for every a in L. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol. 32 No. 2s (2024) 541 https://internationalpubls.com Proof. Since N is a pure element of M, so aN = N ∧ a1M. Hence (aN : 1M ) = ((N ∧ a1M ) : 1M ) = (N : 1M ) ∧ (a1M : 1M ) = (N : 1M ) ∧ a. We need to show that (aN : 1M ) = a(N : 1M ). Obviously, a(N : 1M ) ≤ (aN : 1M ). Conversely, let x ≤ (aN : 1M ). Then x1M ≤ aN = a(N : 1M )1M. Thus x ≤ a(N : 1M ), and hence (aN : 1M ) ≤ a(N : 1M ). Lemma 2.10. Let M be a faithful multiplication L−module. If a(N : 1M) = a ∧ (N : 1M), for every a of L, then N is multiplication and is idempotent in M. Proof. Assume a(N : 1M) = a ∧ (N : 1M ), for all a of L. Take a = (N : 1M ). Then (N : 1M ) 2 = (N : 1M ) and hence (N : 1M ) is an idempotent element of L. Hence N = (N : 1M )1M = (N : 1M ) 21M = (N : 1M )(N : 1M )1M = (N : 1M)N, and hence N is idempotent in M. To prove that N is multiplication, let K be any element of M. Let a = (K : 1M). Then ((K ∧ N ) : 1M) = (K : 1M ) ∧ (N : 1M ) = (K : 1M )(N : 1M ) ≤ (K : N)(N : 1M), and hence K ∧ N = ((K ∧ N) : 1M )1M ≤ (K : N)(N : 1M )1M ≤ (K : N )N ≤ K ∧ N, so that K ∧ N = (K : N )N and N is multiplication. This completes the proof of the theorem. Theorem 2.11. Let L be a CG-multiplicative lattice and M be a multiplication L-module. For N, K in M and a in L. 4. If a is pure in L and N pure in M, then aN is pure in M. In particular, if a is pure in L, then a1M is a pure element of M. 5. If K is pure in N and N pure in M, then K is pure in M. 6. Let K ∨ N be a multiplication element. If each of K and N is pure in M, then K ∨ N and K ∧ N are pure in M. Proof. 1 :⇒ Let b ∈ L. We show that, b(aN) = aN ∧ b1M. Assume that, L is local multiplicative lattice. Since a is a pure in L, then a = 0L or a = 1L. If a = 0L, then we are through. If a = 1L, then the purity of N implies that b(aN ) = bN = N ∧ b1M = aN ∧ b1M . 2:⇒ Let b ∈ L. Then bK = K ∧ bN and bN = N ∧ b1M and hence, bK = (K ∧ N ) ∧ b1M = K ∧ b1M , since K ≤ N . So K is pure in M . 3 :⇒ Given K and N are pure in M , aK = K ∧ a1M and aN = N ∧ a1M . So aK ∧ aN = (K ∧ N ) ∧ a1M and a(K ∨ N ) = (K ∧ a1M ) ∨ (N ∧ a1M ). Since K ∨ N is multiplication, so a(K ∧ N ) = aK ∧ aN and (K ∨ N ) ∧ a1M = (K ∧ a1M ) ∨ (N ∧ a1M ) and this shows that K ∧ N and K ∨ N are pure elements of M . In the following theorem we give a relation between pure elements, multiplication elements and idempotent elements. Theorem 2.12. Let L be a CG-multiplicative lattice and M be a faithful multiplication L- module such that 1M compact. For N in M, the following are equivalent: 1. N is a pure element of M. 2. N is multiplication and is idempotent in M. 3. (N : 1M ) = a ∧ (N : 1M ), for every a ∈ L. Proof. 1 ⇒ 2 : Assume that N is a pure element of M . Let K be a element of M. We will Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol. 32 No. 2s (2024) 542 https://internationalpubls.com show that, N ∧ K = (K : N )N . Since M is multiplication, K = (K : 1M )1M . Since N is a pure element of M, so we have, (K : N )N = N ∧ (K : N )1M . Now (K : N )N = N ∧ (K : N )1M ≥ N ∧ (K : 1M )1M = N ∧ K ≥ (K : N )N. Hence, we get (K : N )N = K ∧ N. This implies that N is a multiplication in M. Since N is pure element, so we have (N : 1M )N = N ∧ (N : 1M )1M = N. N = (N : 1M )N = (N : 1M )(N : 1M )1M = (N : 1M )21M. Hence, we get (N : 1M )21M = (N : 1M )1M. So we have (N : 1M ) is an idempotent element of L. And hence N is idempotent in M. 2 ⇒ 3 : Assume that N is multiplication and idempotent in M . So (N : 1M ) is an idempotent element, then we have N = (N : 1M )1M = (N : 1M )21M = (N : 1M )(N : 1M )1M = (N : 1M )N. So for any element K of M, we have, (K : N )N = (K : N )(N : 1M )N ≤ (K : 1M )N ≤ (K : N )N , that implies (K : N )N = (K : 1M )N . Since N is multiplication element of M , so for every a of L, a1M ∧ N = (a1M : N )N = (a1M : 1M )N = aN = a1M ∧ (N : 1M )1M . Also aN = a(N : 1M )N = a(N : 1M )1M , so a1M ∧ (N : 1M )1M = a(N : 1M )1M for any a ∈ L, hence a1M ∧ (N : 1M )1M = (a ∧ (N : 1M ))1M . So we have, a(N : 1M ) = a ∧ (N : 1M ). 3 ⇒ 1 : Let a ∈ L. we have (N : 1M )1M ∧ a1M = ((N : 1M ) ∧ a)1M . Since (N : 1M ) ∧ a = a(N : 1M ), implies that N ∧ a1M = (N : 1M )1M ∧ a1M = ((N : 1M ) ∧ a)1M = a(N : 1M )1M = aN . Hence N is a pure element in M. Theorem 2.13. Let L be a CG-multiplicative lattice and M a faithful multiplication L-module. If N is pure in M, then (N : 1M ) is the smallest element a ∈ L, such that N = aN. Proof. Let Λ be the collection of all elements a of L with the property that N = aN . Then N = ∧a∈Λ aN = (∧ a∈Λa)N . It follows that (N : 1M ) = ((∧ a∈Λ a)N : 1M ) = (∧ a∈Λ a)(N : 1M ), and hence (N : 1M ) ≤(∧ a∈Λ a). But N is pure, and hence an idempotent. Thus N= (N : 1M )N, and this means that (N : 1M ) ∈ Λ. So (N : 1M ) is the smallest element of Λ. Let M be a L-module. A proper element P of M is called a prime element of M, if P ≠ 1M and whenever rN ≤ P, for some N ∈ M and r ∈ L, then N ≤ P or r ≤ (P : 1M). The M -radical, rad N, of an element N of M is defined as the meet of all prime elements of M containing N. If a is an element of L, then √𝑎 is defined as the meet of all prime elements of L containing a. If a is a pure (and hence idempotent) element of L, then a = a√𝑎. Lemma 2.14. Let N be a element of an L-module M. Then √(𝑁 ∶ 1𝑀) 1M ≤ radN. Proof. If radN = 1M , the result is clear. Otherwise, if P is any prime element of M which contains N, then (N : 1M)≤(P : 1M). As P is a prime element of M, so (P : 1M) is a prime element of L. Hence √(𝑁 ∶ 1𝑀) ≤ (P : 1M ) and thus √(𝑁 ∶ 1𝑀)1M ≤ (P : 1M ) 1M = P. Since P is an arbitrary element containing N, we have √(𝑁 ∶ 1𝑀)1M ≤ radN. Proposition 2.15. [4] Let L be a multiplicative PG-lattice. Let M be a multiplication L- module and ann(M ) ≤ b for some prime element b ∈ L. If a1M ≤ b1M for some a ∈ L, then a ≤ b or b1M = 1M. Lemma 2.16. Let L be a multiplicative PG-lattice. Let M be a multiplication L-module such that Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol. 32 No. 2s (2024) 543 https://internationalpubls.com 1M compact and ann(M ) ≤ a for prime element a ∈ L. Then a1M is a prime element of M. Proof. Note that a1M ≠ 1M and for b ∈ L and N ∈ M, suppose that bN ≤ a1M . As M is a multiplication L-module, we have N = c1M, for c ∈ L, so bN = b(c1M ) ≤ a1M. Proposition 2.15 implies that bc ≤ a, hence b ≤ a or c ≤ a = (a1M : 1M ), then N = c1M ≤ a1M and the proof is complete. Theorem 2.17. Let L be a multiplicative PG-lattice. Let M be a multiplication L-module such that 1M compact and let B be a element of M. Then radB = √(𝐵 ∶ 1𝑀)1M. Proof. By Lemma 2.14, √(𝐵 ∶ 1𝑀)1M ≤ radB. Since M is a multiplication L-module, radB = (radB : 1M )1M. it suffices then to show that (radB : 1M ) ≤√(𝐵 ∶ 1𝑀). Let a be any prime element such that (B : 1M ) ≤ a. Since a is a prime element containing annM , then a1M is a prime element of M containing B = (B : 1M )1M . Hence, (radB : M )1M = radB ≤ a1M , so that (radB : 1M ) ≤ a. Consequently, (radB : 1M ) ≤ √(𝐵 ∶ 1𝑀) . The next result generalizes the above facts to pure element of multiplication L-module. Proposition 2.18. Let L be a CG-multiplicative lattice and M a faithful multiplication L- module. Let N be a pure element of M. Then 1. N = √(𝑁 ∶ 1𝑀)N, 2. (N : 1M )radN = N = (radN : 1M )N. Proof. 1 :⇒ Let be the collection of all prime elements a of L contains (N : 1M ). Then √(𝑁 ∶ 1𝑀)= ∧ a∈Λa, and so, √(𝑁 ∶ 1𝑀) N= (∧ a∈Λa)N= ∧ a∈ΛaN. For each a ∈ Λ, N = (N : 1M )N ≤ aN ≤ N so that N = aN, and hence N = ∧ a∈Λa N=√(𝑁 ∶ 1𝑀)N. 2 :⇒ It follows from (1), and theorem 2.17, that N = √(𝑁 ∶ 1𝑀) N = √(𝑁 ∶ 1𝑀) (N : 1M ) 1M = (N : 1M )radN . But radN ≤ 1M and M is a multiplication L-module. Thus radN = (radN : 1M ) 1M, and hence (N : 1M )radN = (N : 1M )(radN : 1M ) 1M = (radN : 1M ) N. III. Weakly Pure Element In this section we give basic definition of weakly pure element of multiplication L-module, and prove some results related to weakly pure element. We begin with following definition. Definition 3.1. A proper element N of L-module M is called weakly pure, if aN = N ∧ a1M, for every idempotent element a of L. Lemma 3.2. Let M be a faithful multiplication L−module. If N is a weakly pure element of M, then a(N : 1M ) = a ∧ (N : 1M ), for every idempotent element a of L. Proof. Proof follows by Lemma 2.9. Proposition 3.3. Let M be a faithful multiplication L−module. If N is a weakly pure element of M, then (N : 1M ) is idempotent. Proof. By Lemma 3.2, we have (N : 1M )2 = (N : 1M ) ∧ (N : 1M ) = (N : 1M ). Theorem 3.4. Let M be a faithful multiplication L−module, and N is a weakly pure element Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol. 32 No. 2s (2024) 544 https://internationalpubls.com / of M. Then N is primary element of M if and only if it is weakly primary element of M. Proof. It is enough to show that, if N is weakly primary, then N is primary. Assume that 0M ≠ N is a weakly primary element of M that is not primary. Then by Proposition 3.3, we have N = (N : 1M )1M = (N : 1M )21M = (N : 1M )N = 0M, which is a contradiction. Thus N is primary. Proposition 3.5. Let M be a prime multiplication faithful L-module and 0M ≠ N be a proper weakly pure element of M. Then ann(N : 1M ) = 0L. Proof. For every a ≤ ann(N : 1M ), we have a(N : 1M ) = 0L, hence aN = a(N : 1M )N = 0M , so that a ≤ annN = annM = 0L, since M is prime. Hence a = 0L, so ann(N : 1M ) = 0L. Proposition 3.6. Let L be a Noetherian multiplicative lattice with Jacobson radical r∗,and M a multiplication L-module and N is a weakly pure element of M. Then there is a maximal element r of L such that (N : 1M ) ≰ r. Proof. 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