Received October 29, 2021. Published online January 18, 2023.
Abstract: Let $R$ and $S$ be commutative rings with identity, $J$ be an ideal of $S$, $f \colon R \to S$ be a ring homomorphism, $M$ be an $R$-module, $N$ be an $S$-module, and let $\varphi\colon M \to N$ be an $R$-homomorphism. The amalgamation of $R$ with $S$ along $J$ with respect to $f$ denoted by $R \bowtie^f J$ was introduced by M. D'Anna et al. (2010). Recently, R. El Khalfaoui et al. (2021) introduced a special kind of $(R \bowtie^f J)$-module called the amalgamation of $M$ and $N$ along $J$ with respect to $\varphi$, and denoted by $M \bowtie^{\varphi} JN$. We study some homological properties of the $(R \bowtie^f J)$-module $M \bowtie^{\varphi} JN$. Among other results, we investigate projectivity, flatness, injectivity, Cohen-Macaulayness, and prime property of the $(R \bowtie^f J)$-module $M \bowtie^{\varphi} JN$ in connection to their corresponding properties of the $R$-modules $M$ and $JN$.
Keywords: amalgamation of ring; amalgamation of module; Cohen-Macaulay; injective module; projective(flat) module
Affiliations: Hanieh Shoar, Department of Mathematics, Science and Research Branch, Islamic Azad University, Tehran, Iran, e-mail: hanieh.shoar@srbiau.ac.ir; Maryam Salimi (corresponding author), Department of Mathematics, East Tehran Branch, Islamic Azad University, Tehran, Iran, e-mail: maryamsalimi@ipm.ir; Abolfazl Tehranian, Hamid Rasouli, Department of Mathematics, Science and Research Branch, Islamic Azad University, Tehran, Iran, e-mail: tehranian@srbiau.ac.ir, hrasouli@srbiau.ac.ir; Elham Tavasoli, Department of Mathematics, East Tehran Branch, Islamic Azad University, Tehran, Iran, e-mail: elhamtavasoli@ipm.ir