On 2-absorbing submodules and their Amalgamations

TL;DR

The paper characterizes $2$-absorbing submodules in amalgamation modules $M \bowtie^{\varphi} JN$ by reducing to the corresponding submodules in $M$ and in $\varphi(M)+JN$ via a key isomorphism $ (M \bowtie^{\varphi} JN)/(F \bowtie^{\varphi} JN) \cong M/F $ and the projection $p_{\gamma}$. It proves that $F \bowtie^{\varphi} JN$ is $2$-absorbing in $M \bowtie^{\varphi} JN$ iff $F$ is $2$-absorbing in $M$, and that $\overline{N_2}^{\varphi}$ is $2$-absorbing iff $N_2$ is $2$-absorbing in $\varphi(M)+JN$, yielding a suite of corollaries including reductions to the unamalgamated cases, localization and chain stability, and implications for idealization and duplication constructions. These results provide a structural framework for studying $2$-absorbing submodules under amalgamations and related constructions, with applications to specialization cases and particular module classes such as valuation and domain settings.

Abstract

Let $R_1$ and $R_2$ be commutative rings with $1\neq 0,\;M$ and $N$ be unitary $R_1-$module and $R_2-$module, respectively. $f:R_1\rightarrow R_2$ be a ring homomorphism and $\varphi: M\rightarrow N$ be an $R-$module homomorphism. This article studied $2-$absorbing submodule that is a generalization of the concept of prime submodule. Firstly, some characterizations of $2-$absorbing submodule are presented. Then, we examine the notion of $2-$absorbing submodule in amalgamation module $M \bowtie^{\varphi} JN$. We detected when $F\bowtie^\varphi JN$ is a $2-$absorbing submodule of $M \bowtie^{\varphi} JN$ by using the isomorphism $\frac{M \bowtie^{\varphi} JN}{F \bowtie^{\varphi} JN} \cong \frac{M}{F}$ and the homomorphism $p_γ: M \bowtie^{\varphi} JN \rightarrow \varphi(M)+JN$, where $J$ be an ideal of $R_2$ and $F$ be a submodule of $M.$

Theorems & Definitions (30)

• Proposition 2.1
• proof
• Proposition 2.2
• proof
• Proposition 2.3
• proof
• Proposition 2.4
• proof
• Proposition 2.5
• proof