AMALGAMATED MODULES ALONG AN IDEAL
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Let $R$ and $S$ be two commutative rings, $J$ be an ideal of $S$ and $f:R\to S$ be a ring homomorphism. The amalgamation of $R$ and $S$ along $J$ with respect to $f$, denoted by $R\bowtie^{f}J$, is the special subring of $R\times S$ defined by $R\bowtie^{f}J=\{(a,f(a)+j)\,|\, a\in R, j\in J\}$. In this paper, we study some basic properties of a special kind of $R\bowtie^{f}J$-modules, called the amalgamation of $M$ and $N$ along $J$ with respect to $\varphi$, and defined by $M\bowtie^{\varphi}JN:=\{(m,\varphi(m)+n)\mid m\in M\text{ and }n\in JN\}$, where $\varphi:M\to N$ is an $R$-module homomorphism. The new results generalize some known results on the amalgamation of rings and the duplication of a module along an ideal.
Abstract
Let $R$ and $S$ be two commutative rings, $J$ be an ideal of $S$ and $f:R\to S$ be a ring homomorphism. The amalgamation of $R$ and $S$ along $J$ with respect to $f$, denoted by $R\bowtie^{f}J$, is the special subring of $R\times S$ defined by $R\bowtie^{f}J=\{(a,f(a)+j)\,|\, a\in R, j\in J\}$. In this paper, we study some basic properties of a special kind of $R\bowtie^{f}J$-modules, called the amalgamation of $M$ and $N$ along $J$ with respect to $\varphi$, and defined by $M\bowtie^{\varphi}JN:=\{(m,\varphi(m)+n)\mid m\in M\text{ and }n\in JN\}$, where $\varphi:M\to N$ is an $R$-module homomorphism. The new results generalize some known results on the amalgamation of rings and the duplication of a module along an ideal.
참고문헌 (7)
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