Let $R$ and $S$ be commutative rings with identity, $J$ be an ideal of $S$, $f:R\to S$ be a ring homomorphism, $M$ be an $R$-module, $N$ be an $S$-module, and let $\varphi :M\to N$ 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 $\left(R{\bowtie }^{f}J\right)$-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 $\left(R{\bowtie }^{f}J\right)$-module $M{\bowtie }^{\varphi }JN$. Among other results,...

The aim of this paper is to discuss the flat covers of injective modules over a Noetherian ring. Let R be a commutative Noetherian ring and let E be an injective R-module. We prove that the flat cover of E is isomorphic to ${\prod }_{p\in At{t}_R\left(E\right)}{T}_p$. As a consequence, we give an answer to Xu’s question [10, 4.4.9]: for a prime ideal p, when does $T_p$ appear in the flat cover of E(R/m̲)?