On generalizations of classical primary submodules over commutative rings

Abstract: Let :(M) → (M) ∪ {�} be a function where (M) is the set of all submodules of R-module M. A proper submodule N of M is called a φ-classical primary submodule, if for each m ∊ M and a, b ∊ R with abm ∊ N-φ(N), then am ∊ N or bnm ∊ N for some positive integer n. Some characterizations of classical primary and φ-classical primary submodules are obtained. It is shown that N is a φ-classical primary submodule of M if and only if for every m ∊ M−N, (N:m) is a φ-primary ideal of R where (φ(N):m) = φ(N:m). Moreover, we investigate relationships between classical primary, φ-classical primary and φ-primary submodules of modules over commutative rings. Finally, we obtain necessary and sufficient conditions of a φ-classical primary submodule in order to be a φ-primary submodule.


Introduction
Throughout this paper, we assume that all rings are commutative with 1 ≠ 0. Let R be a commutative ring and M be an R-module. We will denote by (N:M) the residual of N by M, that is, the set of all r ∊ R such that rM ⊆ N. Let I be a proper ideal of R. Then √ I = {r ∈ R:r n ∈ I, for some positive integer n} denotes the radical ideal of R. A proper ideal I of R is called a weakly primary ideal if whenever 0 ≠ ab ∊ I for a, b ∊ R, then a ∊ I or b ∈ √ I. The notion of weakly primary ideals has been introduced and studied by Atani and Farzalipour (2005). Anderson and Badawi (2011) generalized the concept of 2-absorbing ideals to n-absorbing ideals. According to their definition, a proper ideal I of R is said to be an n-absorbing ideal of R if whenever a 1 a 2 … a n+1 ∊ I for a 1 , a 2 , …, a n+1 ∊ R, then there are n of the a i 's whose product is in I. Later, Badawi, Tekir, and Yetkin (2015) generalized the concept of weakly primary ideals to weakly 2-absorbing primary ideals. According to their definition, a proper ideal I of R is said to be a weakly 2-absorbing primary ideal of R if whenever 0 ≠ abc ∊ I for a, b, c ∊ R, then ab ∊ I or ac ∈ √ I or bc ∈ √ I. Clearly, every weakly primary ideal is a weakly 2-absorbing primary ideal.
Also, Tekir, Koc, and Oral (2016) generalized the concept of quasi-primary ideals to 2-absorbing quasi-primary ideals. According to their definition, a proper ideal I of R is said to be a 2-absorbing quasi-primary ideal of R if √ I is a 2-absorbing ideal of R. Thus, a 2-absorbing quasi-primary ideal is quasi-primary.
Let :(R) → (R) ∪ {�} be a function where (R) is a set of ideals of R. A proper ideal I of R is called a ϕ-prime ideal of R as in Anderson and Bataineh (2008) if whenever ab ∊ I − ϕ(I) for a, b ∊ R, then a ∊ I or b ∊ I. Darani (2012) generalized the concept of primary and weakly primary ideals to ϕ-primary ideals. A proper ideal I of R is said to be a ϕ-primary ideal of R if whenever ab ∊ I − ϕ(I) for a, b ∊ R, then a ∊ I or b ∈ √ I. Clearly, every ϕ-prime ideal is a ϕ-primary ideal. Later, Badawi, Tekir, Ugurlu, Ulucak, and Celikel (2016) generalized the concept of 2-absorbing primary ideals to ϕ-2absorbing primary ideals. According to their definition, a proper ideal I of R is said to be a ϕ-2- I. Thus, a ϕ-primary ideal is ϕ-2-absorbing primary.
In 2004, Behboodi introduced the concepts of a classical prime submodule. A proper submodule N of an R-module M is said to be a classical prime submodule of M if whenever abm ∊ N for a, b ∊ R, m ∊ M, then am ∊ N or bm ∊ N. (see also Azizi, 2006;Azizi, 2008;Behboodi, 2006, in which, the notion of classical prime submodules is named "weakly prime submodules"). For more information on classical prime submodules, the reader is referred to (Arabi-Kakavand & Behboodi, 2014;Behboodi, 2007;Behboodi & Shojaee, 2010;Yılmaz & Cansu, 2014). Later, Baziar and Behboodi (2009) introduced the concepts of a classical primary submodule. According to their definition, a proper submodule N of M is said to be a classical primary submodule of M if whenever abm ∊ N for a, b ∊ R, m ∊ M, then am ∊ N or b n m ∊ M for some positive integer n. Clearly, every classical prime submodule is a classical primary. Also, Behboodi, Jahani-Nezhad, and Naderi (2011) introduced the concepts of a classical quasi-primary submodule. According to their definition, a proper submodule N of M is said to be a classical quasi-primary submodule of M if whenever abm ∊ N for a, b ∊ R, m ∊ M, then a n m ∊ N or b n m ∊ N for some positive integer n. Thus, a classical primary submodule is classical quasi-primary. The notion of weakly classical primary submodules has been introduced and studied by Mostafanasab (2015). A proper submodule N of an R-module M is said to be a weakly classical primary submodule of M if whenever 0 ≠ abm ∊ N for a, b ∊ R, m ∊ M, then am ∊ N or b n m ∊ N for some positive integer n. Mostafanasab, Tekir, and Oral (2016) introduced the concepts of a weakly classical prime submodule. According to their definition, a proper submodule N of M is said to be a weakly classical prime submodule of M if whenever 0 ≠ abm ∊ N for a, b ∊ R, m ∊ M, then am ∊ N or bm ∊ N. Zamani (2010), generalized the concept of prime and weakly prime submodules to ϕ-prime sub- . Moreover, we investigate relationships between classical primary, ϕ-classical primary and ϕ-primary submodules of modules over commutative rings. Finally, we obtain necessary and sufficient conditions of a ϕ-classical primary submodule in order to be a ϕ-primary submodule.

Some basic properties of ϕ-classical primary submodules
The results of the following theorems seem to play an important role to study ϕ-classical primary submodules of modules over commutative rings; these facts will be used frequently and normally, we shall make no reference to this definition. Remark 2.2. It is easy to see that every classical primary submodule is ϕ-classical primary.
The following example shows that the converse of Remark 2.2 is not true.
Throughout the rest of this paper, M is an R-module and : out loss of generality, we will assume that ϕ(N) ⊆ N and ϕ(I) ⊆ I.Theorem 2.4.Let M be an R-module. Then, the following statements hold: ( The following example shows that the converse of Theorem 2.4 is not true. , but 2(0, 1, 1) ∉ N and 3 n (0, 1, 1) ∉ N for all positive integer n. Therefore, N is not a ϕ-classical primary submodule of M. for all s ∊ S (Larsen & McCarthy, 1971). We know that every submodule of S −1 M is of the form S −1 N for some submodule N of M (Sharp, 2000). (1) N is a ϕ-classical primary submodule of M.

Let
(2) For every a ∊ R and m ∊ M if a n m ∉ N for all positive integer n, then (N:am) = (ϕ(N):am) ∪ (N:m).
( (1) N is a ϕ-classical primary submodule of M.
(2) For every a ∊ R, m ∊ M and every ideal I of R if aIm ⊆ N − ϕ(N), then Im ⊆ N or a n m ∊ N for some positive integer n.
( ideals I 1 and I 2 of R. A product of N 1 and N 2 denoted by N 1 N 2 is defined by N 1 N 2 = I 1 I 2 M. The following theorem offers a characterization of ϕ-classical primary submodules.
Theorem 3.7 Let R be a noetherian ring and let N be a proper submodule of a multiplication R-module M. Then the following conditions are equivalent: (1) N is a ϕ-classical primary submodule of M.
(2) If K 1 K 2 ⊆ N − ϕ(N) for some submodules K 1 , K 2 of M, then K 1 ⊆ N or K n 2 ⊆ N for some positive integer n.
Proof (1 ⇒ 2) Suppose that K 1 , K 2 are submodules of M. Since M is multiplication, there are ideals I 1 , I 2 of R such that K 1 = I 1 M and K 2 = I 2 M. Let m ∊ M. Then I 1 I 2 m ⊆ I 1 I 2 M = K 1 K 2 ⊆ N − ϕ(N). By Theorem 3.6, i.e. I 1 m ⊆ N or I 2 ⊆ √ (N:m). Therefore I 1 M ⊆ N or I n 2 M ⊆ N for some positive integer n. Hence, K 1 ⊆ N or K n 2 ⊆ N for some positive integer n.
(2 ⇒ 1) Let m ∊ M and I 1 mI 2 m = I 1 I 2 m ⊆ N − ϕ(N) for some ideals I 1 , I 2 of R. Thus by part 2, i.e. I 1 m ⊆ N or I n 2 m ⊆ N for some positive integer n. Thus I 1 m ⊆ N or I 2 ⊆ √ (N:m). By Theorem 3.6, N is ϕ-classical primary submodule of M. □ We are finding additional condition to show that a classical primary submodule is a ϕ-classical primary submodule of an R-module M. Remark 3.10 It is easy to see that every ϕ-primary submodule is ϕ-classical primary.
The following example shows that the converse of Remark 3.10 is not true. It is easy to see that N is a ϕ-classical primary submodule of M. Notice that 4(0, 1) ∈ 0 × 4Z, but (0, 1) ∉ 0 × 4Z and 4 ∉ √ ( 0 × 4Z:Z × Z). Therefore N is not a ϕ-primary submodule of M. We provide some relationships between ϕ-classical primary submodules of an R-module M and ϕprimary submodule of M. However, these results require that M be a cyclic R-module. Proof This makes the same assertion as Theorem 3.12. □ Now, we are finding additional condition to show that a ϕ-primary submodule is a ϕ-classical primary submodule of an R-module M.