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Let $(R,𝔪)$ be a commutative Noetherian local ring. We establish some bounds for the sequence of Bass numbers and their dual for a finitely generated $R$-module.

Let $R$ be a commutative ring with identity and $A\left(R\right)$ be the set of ideals with nonzero annihilator. The strongly annihilating-ideal graph of $R$ is defined as the graph $\mathrm{SAG}\left(R\right)$ with the vertex set $A{\left(R\right)}^{*}=A\left(R\right)\setminus \left\{0\right\}$ and two distinct vertices $I$ and $J$ are adjacent if and only if $I\cap \mathrm{Ann}\left(J\right)\ne \left(0\right)$ and $J\cap \mathrm{Ann}\left(I\right)\ne \left(0\right)$. In this paper, the perfectness of $\mathrm{SAG}\left(R\right)$ for some classes of rings $R$ is investigated.

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,...