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,...

We define nice partitions of the multicomplex associated with a Stanley ideal. As the main result we show that if the monomial ideal $I$ is a CM Stanley ideal, then $I^p$ is a Stanley ideal as well, where $I^p$ is the polarization of $I$.