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For commutative rings with identity, we introduce and study the concept of semi $r$-ideals which is a kind of generalization of both $r$-ideals and semiprime ideals. A proper ideal $I$ of a commutative ring $R$ is called semi $r$-ideal if whenever $a^{2}\in I$ and $Ann_{R}(a)=0$, then $a\in I$. Several properties and characterizations of this class of ideals are determined. In particular, we investigate semi $r$-ideal under various contexts of constructions such as direct products, localizations, homomorphic images, idealizations and amalagamations rings. We extend semi $r$-ideals of rings to semi $r$-submodules of modules and clarify some of their properties. Moreover, we define submodules satisfying the $D$-annihilator condition and justify when they are semi $r$-submodules.

Let $R$ be a commutative ring with non-zero identity. In this paper, we introduce the concept of weakly $J$-ideals as a new generalization of $J$-ideals. We call a proper ideal $I$ of a ring $R$ a weakly $J$-ideal if whenever $a,b\in R$ with $0\neq ab\in I$ and $a\notin J(R)$, then $a\in I$. Many of the basic properties and characterizations of this concept are studied. We investigate weakly $J$-ideals under various contexts of constructions such as direct products, localizations, homomorphic images. Moreover, a number of examples and results on weakly $J$-ideals are discussed. Finally, the third section is devoted to the characterizations of these constructions in an amagamated ring along an ideal.

Let $R$ be a commutative ring with identity and $M$ be a unitary $R$-module. The aim of this paper is to extend the notion of quasi $J$-ideals of commutative rings to quasi $J$-submodules of modules. We call a proper submodule $N$ of $M$ a quasi $J$-submodule if whenever $r\in R$ and $m\in M$ such that $rm\in N$ and $r\notin(J(R)M:M)$, then $m\in M$-$rad(N)$. We present various properties and characterizations of this concept (especially in finitely generated faithful multiplication modules). Furthermore, we provide new classes of modules generalizing presimplifiable modules and justify their relation with (quasi) $J$-submodules. Finally, for a submodule $N$ of $M$ and an ideal $I$ of $R$, we characterize the quasi $J$-ideals of the idealization ring $R(+)M$.

Let $R$ be a commutative ring with nonzero identity, and $��:\mathcal{I(R)}\rightarrow\mathcal{I(R)}$ be an ideal expansion where $\mathcal{I(R)}$ the set of all ideals of $R$. In this paper, we introduce the concept of $��$-$n$-ideals which is an extension of $n$-ideals in commutative rings. We call a proper ideal $I$ of $R$ a $��$-$n$-ideal if whenever $a,b\in R$ with$\ ab\in I$ and $a\notin\sqrt{0}$, then $b\in ��(I)$. For example, $��_{1}$ is defined by $��_{1}(I)=\sqrt{I}.$ A number of results and characterizations related to $��$-$n$-ideals are given. Furthermore, we present some results related to quasi $n$-ideals which is for the particular case $��=��_{1}.$

Let R be a commutative ring with identity. In this paper, we introduce the concept 1-absrbing primary ideal of R.

Let R be a commutative ring with unity and M be an R- module In this paper we introduce semi n- absorbing and (k, n)-closed submodules of modules over commutative rings, and investigate their basic properties.