Let $I$ be a semi-prime ideal. Then $P_{\circ }\in Min\left(I\right)$ is called irredundant with respect to $I$ if $I\ne {\bigcap }_{P_{\circ }\ne P\in Min\left(I\right)}P$. If $I$ is the intersection of all irredundant ideals with respect to $I$, it is called a fixed-place ideal. If there are no irredundant ideals with respect to $I$, it is called an anti fixed-place ideal. We show that each semi-prime ideal has a unique representation as an intersection of a fixed-place ideal and an anti fixed-place ideal. We say the point $p\in \beta X$ is a fixed-place point if $O^p\left(X\right)$ is a fixed-place ideal. In this situation...

Let $f:A\to B$ and $g:A\to C$ be two ring homomorphisms and let $J$ and $J^{\text{'}}$ be ideals of $B$ and $C$, respectively, such that $f^{-1}\left(J\right)=g^{-1}\left(J^{\text{'}}\right)$. In this paper, we investigate the transfer of the notions of Gaussian and Prüfer rings to the bi-amalgamation of $A$ with $(B,C)$ along $(J,J^{\text{'}})$ with respect to $(f,g)$ (denoted by $A{\bowtie }^{f,g}(J,J^{\text{'}})),$ introduced and studied by S. Kabbaj, K. Louartiti and M. Tamekkante in 2013. Our results recover well known results on amalgamations in C. A. Finocchiaro (2014) and generate new original examples of rings possessing these properties.

Let $A$ be a commutative ring and $𝒮$ a multiplicative system of ideals. We say that $A$ is $𝒮$-Noetherian, if for each ideal $Q$ of $A$, there exist $I\in 𝒮$ and a finitely generated ideal $F\subseteq Q$ such that $IQ\subseteq F$. In this paper, we study the transfer of this property to the polynomial ring and Nagata’s idealization.

Let $R$ and $S$ be commutative rings with unity, $f:R\to S$ a ring homomorphism and $J$ an ideal of $S$. Then the subring $R{\bowtie }^{f}J:=\{(a,f\left(a\right)+j)\mid a\in R$ and $j\in J\}$ of $R\times S$ is called the amalgamation of $R$ with $S$ along $J$ with respect to $f$. In this paper, we determine when $R{\bowtie }^{f}J$ is a (generalized) filter ring.