It is well known that an integral domain is a valuation domain if and only if it possesses only one finitary ideal system (Lorenzen $r$-system of finite character). We prove an analogous result for root-closed (cancellative) monoids and apply it to give several new characterizations of Prüfer (multiplication) monoids and integral domains.

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 $S$...

Let $𝒵\left(ℛ\right)$ be the set of zero divisor elements of a commutative ring $R$ with identity and $ℳ$ be the space of minimal prime ideals of $R$ with Zariski topology. An ideal $I$ of $R$ is called strongly dense ideal or briefly $sd$-ideal if $I\subseteq 𝒵\left(ℛ\right)$ and $I$ is contained in no minimal prime ideal. We denote by $R_K\left(ℳ\right)$, the set of all $a\in R$ for which $\overline{D\left(a\right)}=\overline{ℳ\setminus V\left(a\right)}$ is compact. We show that $R$ has property $\left(A\right)$ and $ℳ$ is compact if and only if $R$ has no $sd$-ideal. It is proved that $R_K\left(ℳ\right)$ is an essential ideal (resp., $sd$-ideal) if and only if $ℳ$ is an almost locally compact...

Let $R$ be a one-dimensional analytically irreducible ring and let $I$ be an integral ideal of $R$. We study the relation between the irreducibility of the ideal $I$ in $R$ and the irreducibility of the corresponding semigroup ideal $v\left(I\right)$. It turns out that if $v\left(I\right)$ is irreducible, then $I$ is irreducible, but the converse does not hold in general. We collect some known results taken from [5], [4], [3] to obtain this result, which is new. We finally give an algorithm to compute the components of an irredundant decomposition...