� EXTRACTA MATHEMATICAE Volumen 33, Número 2, 2018 instituto de investigación de matemáticas de la universidad de extremadura EXTRACTA MATHEMATICAE Vol. 35, Num. 1 (2020), 43 – 54 doi:10.17398/2605-5686.35.1.43 Available online January 29, 2020 A remark on prime ideals S.C. Lee 1, R. Varmazyar 2,@ 1 Department of Mathematics Education and Institute of Pure and Applied Mathematics Jeonbuk National University, Jeonju, Jeonbuk 54896, South Korea 2 Department of Mathematics, Khoy Branch, Islamic Azad University Khoy 58168-44799, Iran scl@jbnu.ac.kr , varmazyar@iaukhoy.ac.ir Received September 25, 2019 Presented by Consuelo Mart́ınez Accepted January 14, 2020 Abstract: If M is a torsion-free module over an integral domain, then we show that for each sub- module N of M the envelope EM (N) of N in M is an essential extension of N. In particular, if N is divisible then EM (N) = N. The last condition says that N is a semiprime submodule of M if N is proper. Let M be a module over a ring R such that for any ideals a, b of R, (a ∩ b)M = aM ∩ bM. If N is an irreducible and weakly semiprime submodule of M, then we prove that (N :R M) is a prime ideal of R. As a result, we obtain that if p is an irreducible ideal of a ring R such that a2 ⊆ p (a is an ideal of R) ⇒ a ⊆ p, then p is a prime ideal. Key words: Prime ideal; generalized prime submodule; semiprime submodule; weakly semiprime submodule. AMS Subject Class. (2010): 13C05, 18E40, 13B30, 16D60, 13B25. 1. Introduction and preliminaries Throughout this paper, all rings are commutative with identity and all modules are unitary. For definitions, examples, of prime, weakly prime, semiprime, and weakly semiprime submodules of R-modules, and relations among them we refer the reader to [8]. In this paper, we do not deal with weakly prime, but we deal with prime, semiprime, and weakly semiprime submodules of R-modules. More precisely, let’s consider the three conditions, for a submodule N of an R- module M, given as follows: 1. For a ∈ R, m ∈ M with am ∈ N, either a ∈ (N :R M) or m ∈ N. 2. For each a ∈ R, m ∈ M with a2m ∈ N, am ∈ N. 3. For each a ∈ R with a2M ⊆ N, aM ⊆ N. @ Corresponding author ISSN: 0213-8743 (print), 2605-5686 (online) https://doi.org/10.17398/2605-5686.35.1.43 mailto:scl@jbnu.ac.kr mailto:varmazyar@iaukhoy.ac.ir mailto:varmazyar@iaukhoy.ac.ir https://www.eweb.unex.es/eweb/extracta/ https://creativecommons.org/licenses/by-nc/3.0/ 44 s.c. lee, r. varmazyar A proper submodule N of an R-module M is said to be prime, semiprime, and weakly semiprime if it satisfies the condition (1), the condition (2), and the condition (3), respectively. We can summarize the results as follows: prime ⇒ : semiprime ⇒ : weakly semiprime Moreover, if N is a prime submodule of an R-module M then it follows from [11, Proposition 2.4] that (N : M) is a prime ideal of R. By [11, Proposition 2.4], every prime submodule is semiprime and by [4, Theorem 11] there are some semiprime submodules that are not prime. It was shown in [8] that every semiprime submodule is weakly semiprime and the converse is not true in general. The concept of prime number in the ring of integers is generalized to the one of prime ideal of a ring. In turn, the concept of prime ideal of a ring is generalized to the one of prime submodule of a module. Several authors have studied the subject for a couple of decades and many results have been given to explore the nature of prime ideals and related concepts. Section 2 is devoted to primes and generalized primes. If M is a finitely generated multiplication module over a ring R, then it is shown that for any submodule N of M, radM (N) = √ (N :R M)M (Theorem 2.1). Let R be a ring. Let M be an R-module. Assume that M is a multiplication R-module, and that for any ideals a, b of R, (P :M b) ∩ (P :M a) ⊆ P . Then we show that P is a generalized prime submodule of an R-module M if and only if (P :R M) is a prime ideal of R (Theorem 2.2). In Section 3, we concerned about semiprime submodules. Let M be a torsion-free module, or a projective module over an integral domain R. Then we show that for every submodule N of M, EM (N) is an essential extension of N, and in particular, for every divisible submodule N of M, EM (N) = N and EM (N) is the injective envelope of N (Theorem 3.1, Corollary 3.1). In Section 4, we deal with weakly semiprimes. If M is a module over a ring R such that for any ideals a, b of R, (a ∩ b)M = aM ∩ bM and if N is an irreducible and weakly semiprime submodule of M, then we prove that (N :R M) is a prime ideal of R (Theorem 4.1). In particular, if p is an irreducible ideal of a ring R such that a2 ⊆ p (a is an ideal of R) ⇒ a ⊆ p, then we show that p is a prime ideal of R (Corollary 4.2). a remark on prime ideals 45 Notations: Let Spec(M) be the spectrum of prime submodules of an R-module M. For a submodule N of an R-module M and an ideal a of a ring R, we write (N :R M) = {a ∈ R : aM ⊆ N}, (N :M a) = {x ∈ M : rx ∈ N for all r ∈ a}, V (N) = {P ∈ Spec(M) : P ⊇ N}, √ a = {a ∈ R : an ∈ a for some positive integer n}. For undefined terminologies and notations, we refer the reader to references [10, 6]. 2. Primes and generalized primes This section is dedicated to the relationship between primes and general- ized primes. If R is a ring and a is an ideal of R, then it is well-known that the radical√ a of a is an ideal of R (see, for example, [2, p. 33]), and √ a = ∩p∈V (a)p (see, for example, [2, Corollary 2.12]). Definition 2.1. Let N be a submodule of an R-module M. Then the radical of N in M is defined to be the intersection of prime submodules of M containing N, and is denoted by radM (N). That is, radM (N) = ∩P∈V (N)P . Remark 2.1. For any R-module M, radM (M) = M because V (M) = ∅. Definition 2.2. For any R-module M, the radical of M is defined to be the radical of the zero submodule in M and is denoted by rad(M). That is, rad(M) = radM (0) = ∩P∈Spec(M)P. An ideal a of R is called a radical ideal if a = √ a (see [2, p. 33]). Definition 2.3. A submodule N of an R-module M is called a radical submodule of M if radM (N) = N. Every module is the radical submodule of itself. An R-module M is called a multiplication module if for each submodule N of M, N = (N :R M)M (see [1, 3]). 46 s.c. lee, r. varmazyar Theorem 2.1. Let M be a finitely generated multiplication module over a ring R. Then for any submodule N of M, radM (N) = ∩p∈V (N:RM)(pM) = (∩p∈V (N:RM)p)M = √ (N :R M)M. Proof. A map ϕ : Spec(R/(N :R M)) → Spec(M/N) defined by ϕ(p/(N :R M)) = (pM)/N is an order-preserving 1:1 correspondence (see [3]), so that the first equality follows. The second one follows from the fact that M is a multiplication module, and the third one follows from the fact in the first paragraph of this section. Corollary 2.1. ([2, Corollary 2.12]) If R is a ring and a is an ideal of R, then √ a = ∩p∈V (a)p. Let R be a ring, then every prime ideal of R has the condition: a∩b ⊆ p (a,b are ideals of R) ⇒ a ⊆ p or b ⊆ p. However, a prime submodule P of M does not satisfy the following condition in general: A∩B ⊆ P (A,B are submodules of M) ⇒ either A ⊆ P or B ⊆ P. (2.1) An example of this is given below. Example 2.1. Let R be an integral domain. Then R2( = R ⊕ R) is a torsion-free R-module, so that the submodule 〈(0, 0)〉 is a prime submodule. Moreover, (R × 〈0〉) ∩ (〈0〉 × R) = 〈(0, 0)〉, but R × 〈0〉 6= 〈(0, 0)〉, and 〈0〉×R 6= 〈(0, 0)〉. This shows that the zero submodule of R2 does not satisfies the condition (2.1). A proper submodule P of an R-module M is called generalized prime if P satisfies condition (2.1). Lemma 2.1. Assume that M is a multiplication R-module. If P is a prime submodule of an R-module M, then P is a generalized prime sub- module of M. a remark on prime ideals 47 Proof. Assume that A∩B ⊆ P . Let a, b be ideals of R such that A = aM and B = bM. Then (ab)M ⊆ aM∩bM = A∩B ⊆ P , so that ab ⊆ (P :R M). Notice that (P :R M) is a prime ideal of R. Then either a ⊆ (P :R M) or b ⊆ (P :R M). Hence either aM ⊆ P or bM ⊆ P , that is, either A ⊆ P or B ⊆ P . Therefore, P is generalized prime. Corollary 2.2. Let R be a ring. Then every prime ideal of R is gener- alized prime. Not every generalized prime ideal of a ring R is prime. An example of this is given below. Example 2.2. Let p be a prime number. Consider the ring Zp2 = {0, 1, 2, . . . , p 2 − 1}. Then its only ideals are 〈0〉, 〈p〉, and Zp2 . Hence it is easy to see that the zero ideal 〈0〉 is not prime, but generalized prime. Lemma 2.2. Assume that for any ideals a, b of R, (P :M b) ∩ (P :M a) ⊆ P. If P is a generalized prime submodule of an R-module M, then (P :R M) is a prime ideal of R. Proof. Let ab ⊆ (P :R M). Then (ab)M ⊆ P , so that aM ⊆ (P :M b) and bM ⊆ (P :M a). By our hypothesis aM ∩ bM ⊆ P . Since P is generalized prime, either aM ⊆ P or bM ⊆ P . So, either a ⊆ (P :R M) or b ⊆ (P :R M). Therefore (P :R M) is a prime ideal of R. Corollary 2.3. Let R be a ring. Assume that for any ideals a, b of R, (p :R b) ∩ (p :R a) ⊆ p. If p is a generalized prime ideal of R, then p is a prime ideal of R. Let’s summarize the results as follows: Theorem 2.2. Let R be a ring. Let M be an R-module. Assume that M is a multiplication R-module, and that for any ideals a, b of R, (P :M b) ∩ (P :M a) ⊆ P. Then P is a generalized prime submodule of M if and only if (P :R M) is a prime ideal of R. 48 s.c. lee, r. varmazyar Proof. (⇒) : This follows from Lemma 2.2. (⇐) : This follows from [3, Corollary 2.11] and Lemma 2.1. Corollary 2.4. Let R be a ring. Assume that for any ideals a, b of R, (p :R b) ∩ (p :R a) ⊆ p. Then p is a prime ideal of R if and only if p is a generalized prime ideal of R. 3. Semiprimes In this section, we consider what conditions can be given to an R-module M and to its submodule N, in order for a submodule N of an R-module M to be a semiprime submodule. Let M be an R-module, and let N be a submodule of M. Then recall that the envelope of N in M is defined by EM (N) := 〈 ∪r∈R ( rM ∩ (N :M r) )〉 = 〈 GM (N) 〉 (see [8, p. 3744]). For every submodule N of an R-module M, N ⊆ EM (N) ⊆ M. In partic- ular, EM (M) = M for every R-module M. Lemma 3.1. For every proper submodule N of an R-module M, the fol- lowing assertions are equivalent: (a) EM (N) = N. (b) N is semiprime. (c) For each r ∈ R, m ∈ M with rkm ∈ N for some positive integer k, rm ∈ N. Proof. (a) ⇒ (b) : Let r ∈ R, m ∈ M with r2m ∈ N. Then rm ∈ rM ∩ (N :R r) ⊆∪a∈R(aM ∩ (N :R a)) ⊆ EM (N) = N. (b) ⇒ (a) : To show that EM (N) = N, it suffices to prove that ∪r∈R(rM∩ (N :M r)) ⊆ N. Let x ∈ ∪r∈R(rM ∩ (N :M r)). Then there exist r ∈ R, m ∈ M such that x = rm and x ∈ (N :R r). So, r2m = rx ∈ N. By assumption, x = rm ∈ N. (b) ⇒ (c) : Let r ∈ R and m ∈ M with rkm ∈ N for some positive integer k. If k ≥ 3, then r2(rk−2m) = rkm ∈ N; hence by (b) rk−1m = r(rk−2m) ∈ N. By proceeding this way, we can get r2m ∈ N. Hence by (b) again we get rm ∈ N. (c) ⇒ (b) is obvious. a remark on prime ideals 49 Example 3.1. Consider the ring Z of integers. Let n > 1. Then nZ is a semiprime ideal of Z if and only if n is a product of distinct prime numbers. For example, 12Z is not semiprime (for 22 · 3 ∈ 12Z, but 2 · 3 /∈ 12Z), but 6Z is semiprime. If E is divisible, then not every submodule of E is divisible. For example Q is divisible as a Z-module but its submodule Z is not. If M is a torsion-free module over an integral domain R, then every sub- module of M is also torsion-free. It is well-known [10] that every R-module M has an injective envelope E(M), but not necessarily unique. (see [12] for examples). It is fairly well-known that every nonsingular module M over R has the unique injective envelope E(M) (see [7, Lemma 2]). Every torsion-free module M over an integral domain R is nonsingular, so that its injective envelope E(M) is unique. Theorem 3.1. Let M be a torsion-free module over an integral domain R. Then the following statements are true: (1) For every submodule N of M, EM (N) is an essential extension of N. (2) For every divisible submodule N of M, EM (N) = N and EM (N) is the injective envelope of N. Proof. (1) Let N be any submodule of M, and let U be a nonzero submodule of EM (N). We can take a nonzero element x ∈ U. Then there are a1, . . . ,an ∈ R and x1, . . . ,xn ∈ GM (N) such that x = a1x1 + · · ·+ anxn. We may assume that all xi’s are nonzero. For each i ∈ Zn, there is ri ∈ R such that xi ∈ riM ∩(N :M ri). Moreover, all ri’s are nonzero and rixi ∈ N. Since M is torsion-free over an integral domain R, (r1 · · ·rn)x 6= 0 and (r1 · · ·rn)x = (a1r2 · · ·rn)(r1x1) + · · · + (anr1 · · ·rn−1)(rnxn) ∈ U ∩N. Hence N is essential in EM (N). (2) Let N be any divisible submodule of M. Then N is torsion-free divisible. Hence it follows from [10, Proposition 2.7] that N is injective. Hence by (1) and [10, Theorem 2.17], EM (N) = N. Moreover, notice that EM (N) is also injective. Then by (1) EM (N) is the injective envelope of N. In Theorem 3.1 (2), the condition “divisible” cannot be deleted. To see this let R = Z, M = Q, and N = Z. Then it is not difficult to see that EM (N) = Q 6= N. 50 s.c. lee, r. varmazyar By [5, Exercise 9G, Section 9, Chapter I], every projective module over an integral domain R is torsion-free. Corollary 3.1. Let M be a projective module over an integral domain R. Then (1) and (2) in Theorem 3.1 hold. Compare the following Corollary 3.2 (3) with [10, Theorem 2.8]. Corollary 3.2. Let R be an integral domain. Then the following asser- tions are true: (1) Every torsion-free divisible R-module is injective ([10, Proposition 2.7]). (2) For every ideal a of R, ER(a) is an essential extension of a. (3) Let a be an ideal of R. Then a is injective if and only if it is divisible. Corollary 3.3. Let M be a torsion free module over an integral domain R. Then every injective submodule N of M is semiprime and EM (N) is the injective envelope of N. Proof. By [10, Proposition 2.6], N is divisible. Corollary 3.4. Let R be an integral domain. Then every divisible ideal of R is semiprime and injective. 4. Weakly semiprimes In this section, we will consider weakly semiprimes. Lemma 4.1. For each proper submodule N of an R-module M the follow- ing assertions are equivalent: (a) N is weakly semiprime. (b) For each ideal a of R with a2M ⊆ N, aM ⊆ N. (c) For each ideal a of R with a2 ⊆ (N :R M), a ⊆ (N :R M). (d) (N :R M) is a semiprime ideal of R. Proof. (a) ⇒ (b) : Let N be any weakly semiprime submodule of M, and let a be any ideal of R with a2M ⊆ N. Let x ∈ aM. Then there exist a1, . . . ,an ∈ a and x1, . . . ,xn ∈ M such that x = a1x1 + · · · + anxn. For a remark on prime ideals 51 each i ∈ Zn, a2iM ⊆ a 2M ⊆ N. Since N is weakly semiprime, aiM ⊆ N. In particular, aixi ∈ N. Hence x = a1x1 + · · · + anxn ∈ N. This shows that aM ⊆ N. The rest of the proof can be proved easily. Corollary 4.1. If every proper ideal of a ring R is semiprime, then every proper submodule of a module over the ring R is weakly semiprime. Proof. Let P be a proper submodule of a module M over the ring R. Assume that a is any ideal of R with a2M ⊆ P . Then a2 ⊆ (P :R M). Moreover, (P :R M) is a proper ideal of R, so by hypothesis (P :R M) is semiprime. Hence a ⊆ (P :R M), so that aM ⊆ P . By Lemma 4.1, P is weakly semiprime. If P is a weakly semiprime submodule of an R-module M, then it is clear that (P :R M) is a proper radical ideal of R, that is, √ (P :R M) = (P :R M). It is known (see [9, Corollary 2.7] ) that, over a commutative ring whose proper ideals are semiprime, every proper submodule of a nonzero module is intersection of prime submodules. Let R be a Dedekind domain, and let a be a nonzero proper ideal of R. It is well-known [2, Corollary 11.9] that a is a finite product of (not necessarily distinct) prime ideals. Hence a is semiprime if only if a is a finite product of distinct prime ideals. Definition 4.1. Let P be a submodule of an R-module M. Then the residue class R-module M/P is called a weakly semiprime module if P is weakly semiprime. Lemma 4.2. For each submodule P of an R-module M, the following assertions are equivalent: (a) The residue class R-module M/P is weakly semiprime. (b) For each ideal a of R with a2(M/P ) = 0, a(M/P ) = 0. (c) For each ideal a of R with aM ⊆ (P :M a), aM ⊆ P . (d) For each ideal a of R with aM + (P :M a) = (P :M a), aM ⊆ P . (e) For each ideal a of R with a(M/(P :M a)) = 0, M/(P :M a) = 0. An R-module M is called meet irreducible or simply, irreducible if M 6= 0 and the intersection of two nonzero submodules of M is always nonzero. A 52 s.c. lee, r. varmazyar submodule N of an R-module M is called irreducible if the quotient R-module M/N is irreducible. Let M be an R-module. Then for any ideals a, b of R (a∩b)M ⊆ aM ∩bM. However, the converse is not true in general. If a submodule N of an R-module M is weakly semiprime, then by Lemma 4.1, (N :R M) is semiprime. In order (N :R M) to be prime, we need more conditions on N as in the following results: Theorem 4.1. Let M be a module over a ring R such that for any ideals a, b of R, (a∩b)M = aM ∩bM. If N is an irreducible and weakly semiprime submodule of M, then (N :R M) is a prime ideal of R. Proof. Let us write p = (N :R M). Clearly, p 6= R. Suppose that p is not prime. Then there exist a,b ∈ R such that ab ∈ p, a /∈ p, b /∈ p. Then aM * N, so that there exists m ∈ M such that am /∈ N. This implies that a(m + N) 6= 0 + N in M/N, so that (p + Ra)(M/N) 6= 0 and by the same argument (p + Rb)(M/N) 6= 0. Hence (p + Ra)(M/N) ∩ (p + Rb)(M/N) 6= 0. However, ((p + Ra) ∩ (p + Rb))2(M/N) ⊆ (p + Ra)(p + Rb)(M/N) ⊆ p(M/N) = 0. Since N is weakly semiprime, it follows from Lemma 4.2 that ((p + Ra) ∩ (p + Rb))(M/N) = 0, so that (p + Ra)(M/P) ∩ (p + Rb)(M/N) = 0. This contradiction shows that p is prime. Corollary 4.2. Let R be a ring and p be an ideal of R. If p is an irreducible ideal of R such that z2 ∈ p (z ∈ R) implies z ∈ p, then p is a prime ideal of R. The following lemma is the statement just prior to [3, Theorem 1.6], but we include and prove here to use it in our classification and to get another result. Lemma 4.3. If M is a projective R-module, then (∩λ∈Λaλ)M = ∩λ∈Λ(aλM) for any nonempy collection of ideals aλ (λ ∈ Λ) of R. a remark on prime ideals 53 Proof. Let F be an R-free module and N be a submodule of F such that F = M ⊕ N. It can be easily proven that (∩λ∈Λaλ)F = ∩λ∈Λ(aλF) for any nonempy collection of ideals aλ ( λ ∈ Λ) of R. This implies that (∩λ∈Λaλ)M ⊕ (∩λ∈Λaλ)N = ∩λ∈Λ(aλ(M ⊕N)) = ∩λ∈Λ(aλM ⊕aλN). (4.1) It follows that (∩λ∈Λaλ)M = ∩λ∈Λ(aλM) and (∩λ∈Λaλ)N = ∩λ∈Λ(aλN). For otherwise, we can take x ∈∩λ∈Λ(aλM)\(∩λ∈Λaλ)M and y ∈∩λ∈Λ(aλN)\(∩λ∈Λaλ)N. Then by (4.1) x+y ∈ (∩λ∈Λaλ)M⊕(∩λ∈Λaλ)N = ∩λ∈Λ(aλM⊕aλN), so that for all λ ∈ Λ, there exist uλ ∈ aλM and vλ ∈ aλN such that x + y = uλ + vλ. Hence x − uλ = vλ − y ∈ M ∩ N = 0, so that x = uλ and y = vλ. Hence x ∈∩λ∈Λ(aλM) and y ∈∩λ∈Λ(aλN). This is a contradiction. If M is a faithful multiplication module, then we can get the same result as in Lemma 4.3 (see [3, Theorem 1.6]). Corollary 4.3. 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Algebra 219 (6) (2015), 2099 – 2123. Introduction and preliminaries Primes and generalized primes Semiprimes Weakly semiprimes