[Paper Review] $S$-secondary submodules of a module

F. Farshadifar|arXiv (Cornell University)|Mar 1, 2020

Rings, Modules, and Algebras6 references4 citations

TL;DR

This paper introduces and investigates $S$-secondary submodules of modules over a commutative ring $R$, generalizing both secondary and $S$-second submodules. It establishes equivalent characterizations of $S$-secondary submodules via annihilator conditions and $s$-powers, proving that under finitely generated and comultiplication assumptions, $S$-secondary and $S$-primary submodules yield the equality $Z_R(M) = W_R(M) = \sqrt{\text{Ann}_R(M)}$. The key contribution is a unified framework linking $S$-secondary structure to annihilator and zero divisor behavior in modules.

ABSTRACT

Let R be a commutative ring with identity, S be a multiplicatively closed subset of R, and let M be an R-module. The aim of this paper is to introduce the notion of S-secondary submodules of M as a generalization of secondary submodules of M and investigate some properties of this class of submodules.

Motivation & Objective

• To generalize the concept of secondary submodules by introducing $S$-secondary submodules for an $R$-module $M$ with respect to a multiplicatively closed subset $S \subseteq R$.
• To establish equivalent conditions for a submodule to be $S$-secondary, particularly involving annihilators and $s$-powers.
• To investigate the relationship between $S$-secondary submodules and $S$-primary submodules in finitely generated and comultiplication modules.
• To characterize when the set of zero divisors $Z_R(M)$ and the set $W_R(M)$ of non-surjective multipliers coincide with $\sqrt{\text{Ann}_R(M)}$ under $S$-secondary or $S$-primary assumptions.

Proposed method

• Define $S$-secondary submodules via a fixed $s \in S$ such that for all $r \in R$, either $srN = sN$ or $(sr)^t N = 0$ for some $t \in \mathbb{N}$.
• Use the equivalence of conditions involving submodules $K$, completely irreducible submodules $L$, and ideals $J$ to characterize $S$-secondary submodules.
• Apply localization and annihilator techniques, particularly leveraging $S^{-1}N \subseteq S^{-1}K$ implying $sN \subseteq K$ for some $s \in S$, in the context of finitely generated modules.
• Utilize the comultiplication module property, where every submodule is of the form $IM$ for some ideal $I$, to relate submodules to annihilators.
• Apply results from ideal theory, such as $1 - x \in \text{Ann}_R(M)$ for some $x \in (saM:_{R}M)$, to derive contradictions and prove annihilator equalities.
• Use maximal ideals and the construction of a set $\Omega$ of elements $s_{\mathfrak{M}}$ not in $\mathfrak{M}$ but with $s_{\mathfrak{M}}M \subseteq L$ to show $M = L$ and hence $M$ is cotorsion-free.

Experimental results

Research questions

• RQ1 When is a submodule $N$ of an $R$-module $M$ considered $S$-secondary for a given multiplicatively closed subset $S \subseteq R$?
• RQ2 What are the equivalent algebraic conditions that characterize $S$-secondary submodules, particularly in terms of annihilators and powers of elements?
• RQ3 Under what conditions does the equality $Z_R(M) = W_R(M) = \sqrt{\text{Ann}_R(M)}$ hold for a finitely generated comultiplication module $M$?
• RQ4 How do $S$-secondary and $S$-primary submodules relate in the context of multiplication and comultiplication modules?
• RQ5 Can every non-zero submodule of a finitely generated comultiplication module be $S$-secondary, and what does this imply about the module's zero divisor and annihilator structure?

Key findings

• An $R$-module $M$ is $S$-secondary if and only if there exists $s \in S$ such that for all $r \in R$, either $srN = sN$ or $(sr)^t N = 0$ for some $t \in \mathbb{N}$, which provides a concrete operational criterion.
• Every $S$-second submodule is $S$-secondary, but the converse does not hold in general, as shown by the example of $\mathbb{Z}_4$ as a $\mathbb{Z}$-module with $S = \mathbb{Z} \setminus 2\mathbb{Z}$.
• If $M$ is a finitely generated comultiplication $R$-module with $\sqrt{\text{Ann}_R(M)} \cap S = \emptyset$, then every non-zero submodule being $S$-secondary implies $W_R(M) = \sqrt{\text{Ann}_R(M)}$, linking the non-surjective multiplier set to the annihilator radical.
• If every proper submodule of a finitely generated multiplication $R$-module $M$ is $S$-primary, then $Z_R(M) = \sqrt{\text{Ann}_R(M)}$, showing that the zero divisor set coincides with the radical of the annihilator.
• For a finitely generated multiplication and comultiplication $R$-module $M$ with $\sqrt{\text{Ann}_R(M)} \cap S = \emptyset$, the three conditions—every non-zero submodule is $S$-secondary, every proper submodule is $S$-primary, and $Z_R(M) = W_R(M) = \sqrt{\text{Ann}_R(M)}$—are equivalent.
• The example $\mathbb{Z}_{p^n}$ with $S = \mathbb{Z} \setminus p\mathbb{Z}$ shows that not all non-zero submodules are $S$-secondary and not all proper submodules are $S$-primary, illustrating the necessity of the assumptions in the main theorems.