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Let $G$ be a group. If every nontrivial subgroup of $G$ has a proper supplement, then $G$ is called an $aS$-group. We study some properties of $aS$-groups. For instance, it is shown that a nilpotent group $G$ is an $aS$-group if and only if $G$ is a subdirect product of cyclic groups of prime orders. We prove that if $G$ is an $aS$-group which satisfies the descending chain condition on subgroups, then $G$ is finite. Among other results, we characterize all abelian groups for which every nontrivial quotient group is an $aS$-group....

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 $R$ be a commutative ring with identity. We study the concept of strongly 1-absorbing primary ideals which is a generalization of $n$-ideals and a subclass of $1$-absorbing primary ideals. A proper ideal $I$ of $R$ is called strongly 1-absorbing primary if for all nonunit elements $a,b,c\in R$ such that $abc\in I$, it is either $ab\in I$ or $c\in \sqrt{0}$. Some properties of strongly 1-absorbing primary ideals are studied. Finally, rings $R$ over which every semi-primary ideal is strongly 1-absorbing primary, and rings $R$ over which every strongly...