In this paper, we deal with the study of intermediate domains between a domain $R$ and a domain $T$ such that $T$ is an intersection of localizations of $R$, namely the pair $\left(R,T\right)$. More precisely, we study the pair $\left(R,R_d\right)$ and the pair $\left(R,\tilde{R}\right)$, where $R_d=\cap \left\{R_M\mid M\in \text{Max}\left(R\right),htM=dimR\right\}$ and $\tilde{R}=\cap \left\{R_M\mid M\in \text{Max}\left(R\right),htM\ge 2\right\}$. We prove that, if $R$ is a Jaffard domain, then $\left(R,R_d\left[n\right]\right)$ is a Jaffard pair, which generalize [5, Théorème 1.9]. We also show that if $R$ is an $S$-domain, then $\left(R,\tilde{R}\right)$ is a residually algebraic pair (that is for each intermediate domain $S$ between $R$ and $\tilde{R}$, if $Q$ is a prime ideal of $S$...

Quasi-injective modules over valuation domains are classified by means of complete sets of cardinal invariants.

We describe the isolated points of an arbitrary topological space $(X,\tau )$. If the $\tau$-specialization pre-order on $X$ has enough maximal elements, then a point $x\in X$ is an isolated point in $(X,\tau )$ if and only if $x$ is both an isolated point in the subspaces of $\tau$-kerneled points of $X$ and in the $\tau$-closure of $\left\{x\right\}$ (a special case of this result is proved in Mehrvarz A.A., Samei K., On commutative Gelfand rings, J. Sci. Islam. Repub. Iran 10 (1999), no. 3, 193–196). This result is applied to an arbitrary subspace of the prime...

We give a short overview on the subject of canonical reduction of a pair of bilinear forms, each being symmetric or alternating, making use of the classification of pairs of linear mappings between vector spaces given by J. Dieudonné.

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...

Let $R$ be a commutative ring with non-zero identity. Various properties of multiplication modules are considered. We generalize Ohm’s properties for submodules of a finitely generated faithful multiplication $R$-module (see [8], [12] and [3]).