2-prime submodules of modules

Abstract

Let R be a commutative ring with unity. And let E be a unitary R-module. This paper introduces the notion of 2-prime submodules as a generalized concept of 2-prime ideal, where proper submodule H of module F over a ring R is said to be 2-prime if , for r R and x F implies that or . we prove many properties for this kind of submodules, Let H is a submodule of module F over a ring R then H is a 2-prime submodule if and only if [N ] is a 2-prime submodule of E, where r R. Also, we prove that if F is a non-zero multiplication module, then [K: F] [H: F] for every submodule k of F such that H K. Furthermore, we will study the basic properties of this kind of submodules.

2-Prime Modules

“…Proof: Let E be prime, so (0) is the prime submodule. Thus by [ [1] Remarks and Examples (2.2)] (0) is 2-prime submodule hence E is a 2-prime module. 2.…”

“…Let E be a module over a ring R with identity. In [1] we introduced a 2-prime submodule as a generalization of a 2-prime ideal. A proper submodule of H of module E over a ring R is said to be 2-prime submodule, if rx∈H, where r∈R, x∈E, either x∈H or r 2 ∈[H: E].…”

Weakly 2-Prime Sub-Modules

“…Here we present the idea of a weakly 2-prime sub-module as an extension of a 2-prime submodule, where a valid sub-module N of M is a 2-prime sub-module if whenever 𝑟𝑚 ∈ 𝑁, 𝑟 ∈ 𝑅, 𝑚 ∈ 𝑀, then either 𝑚 ∈ 𝑁 or 𝑟 2 ∈ [𝑁: 𝑀], and vice versa (see [3]).…”

“…It's not necessary for a weakly 2-prime sub-module to be a quasi-prime, for example: a weakly 2-prime sub-module of the 𝑍-module 𝑍 12 it is zero sub-modules. However, is not quasiprime, because (0 ̅ : 𝑧 3 ̅ ) = 4𝑍 is not the prime ideal of Z, (using [3]). Also, quasi-prime need not be a weakly 2-prime sub-module, for example, the Z-module 𝑍⨁𝑍 ̅ , 𝑁 = 2𝑍⨁(0), 𝑁 is a quasiprime sub-module.…”

Closed (St-Closed) Compressible Modules and Closed (St-Closed) Retractable Modules

“…Let 0 ≠ 𝑈 ≤ 𝑆𝑡𝑐 𝑌, then 𝑈 ≤ 𝑐 𝑌,[3], As Y is a closed compressible, then Y can be embedded in St-closed submodule U of Y. Therefore, Y is an Stclosed compressible module.An R-module Y is termed to be fully prime if all proper submodule U of Y is a prime submodule U,[21] Let Y be an R-module . If Y is fully prime, then Y is St-closed compressible if and only if Y is closed compressible.…”