Multiplication Modules

Abstract

Abstract. Let R be a commutative ring with identity and M be a unital R-module. Then M is called a multiplication module provided for every submodule N of M there exists an ideal I of R such that N = IM. Our objective is to investigate properties of prime and semiprime submodules of multiplication modules. Mathematics Subject Classification: 13C05, 13C13Keywords: Multiplication modules, Prime submodules, Semiprime submodules Throughout this paper all rings will be commutative with identity and all modules will be unitary. Let R be a ring and M be a unital R-module. An R-module M is called a multiplication module provided for each submodule N of M there exists an ideal I of R such that N = IM. We say that I is a presantation ideal of N. Clearly, every submodule of M has a presantation ideal if and only if M is a multiplication module. Let N and K be submodules of a multiplication with N = I 1 M and K = I 2 M for some ideals I 1 and I 2 of R.The product N and K denoted by NK is defined by NK = I 1 I 2 M. Then by [6,theorem 3.4], the product of N and K is independent of presentation of N and K. Note that this definition is different from the definition of ordinary ideal multiplication. Indeed, let R = Z be the ring of integers, and let M = 2Z and N = K = 4Z. Then NK is 16Z by the usual definition and is 8Z by the our definition. Moreover, for a, b ∈ M, by ab we mean the product

δ-small submodule and prime modules

“…Definition 3.16: [Singh, 2001] An ideal 𝐽 of the ring 𝑅 is called insertion of factor property (𝐼𝐹𝑃) is 𝑎𝑏 ∈ 𝐽, 𝑎, 𝑏 ∈ 𝑅, so 𝑎𝑅𝑏 ⊆ 𝐽. Therefore a submodule 𝐴 of ℳ is called (𝐼𝐹𝑃) if 𝑎𝑚 ∈ 𝐴, 𝑎 ∈ 𝑅, 𝑚 ∈ ℳ, so 𝑎𝑅𝑚 ⊆ 𝐴 and a module ℳ has (𝐼𝐹𝑃) if the zero submodule has (𝐼𝐹𝑃).…”

“…All rings in this paper are commutative with 1 and all modules with unitary. An 𝑅-module ℳ is called multiplication if every submodule 𝐴 of ℳ, there exists an ideal 𝐽 such that 𝐴 = 𝐽ℳ [Singh, 2001]. The prime ideal was extended to module by several researchers.…”

Mappings Between Module Lattices

“…In other words, M is a multiplication module if and only if for each submodule N of M there exists an ideal B of R such that N = BM . Multiplication modules have been extensively studied (see, for example, [1] - [4], [8] - [11]). Note the following simple fact about multiplication modules that is included for completeness.…”

Multiplication modules over non-commutative rings

“…A ring A is said to be regular (strongly regular) if a ∈ aAa (respectively, a ∈ a 2 A) for each element a ∈ A. If M is a multiplication module over a commutative regular ring A, then all submodules of M are multiplication modules (see [11]). The first main result of the present paper is as follows.…”