Geodesic
Some homological properties of amalgamated modules along an ideal
Czechoslovak Mathematical Journal, Tome 73 (2023) no. 2, pp. 475-486
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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 $JN$.
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 $JN$.
DOI : 10.21136/CMJ.2023.0411-21
Classification : 13A15, 13C10, 13C11, 13C14, 13C15
Keywords: amalgamation of ring; amalgamation of module; Cohen-Macaulay; injective module; projective(flat) module 
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     title = {Some homological properties of amalgamated modules along an ideal},
     journal = {Czechoslovak Mathematical Journal},
     pages = {475--486},
     year = {2023},
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Shoar, Hanieh; Salimi, Maryam; Tehranian, Abolfazl; Rasouli, Hamid; Tavasoli, Elham. Some homological properties of amalgamated modules along an ideal. Czechoslovak Mathematical Journal, Tome 73 (2023) no. 2, pp. 475-486. doi: 10.21136/CMJ.2023.0411-21

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