Let $ℛL$ be the ring of real-valued continuous functions on a frame $L$. The aim of this paper is to study the relation between minimality of ideals $I$ of $ℛL$ and the set of all zero sets in $L$ determined by elements of $I$. To do this, the concepts of coz-disjointness, coz-spatiality and coz-density are introduced. In the case of a coz-dense frame $L$, it is proved that the $f$-ring $ℛL$ is isomorphic to the $f$-ring $C\left(\Sigma L\right)$ of all real continuous functions on the topological space $\Sigma L$. Finally, a one-one correspondence is...

We characterize prime submodules of $R\times R$ for a principal ideal domain $R$ and investigate the primary decomposition of any submodule into primary submodules of $R\times R.$

Let $R$ be a commutative ring with a nonzero identity. In this study, we present a new class of ideals lying properly between the class of $n$-ideals and the class of $(2,n)$-ideals. A proper ideal $I$ of $R$ is said to be a quasi $n$-ideal if $\sqrt{I}$ is an $n$-ideal of $R.$ Many examples and results are given to disclose the relations between this new concept and others that already exist, namely, the $n$-ideals, the quasi primary ideals, the $(2,n)$-ideals and the $pr$-ideals. Moreover, we use the quasi $n$-ideals to characterize some...

We consider analogies between the logically independent properties of strong going-between (SGB) and going-down (GD), as well as analogies between the universalizations of these properties. Transfer results are obtained for the (universally) SGB property relative to pullbacks and Nagata ring constructions. It is shown that if $A\subseteq B$ are domains such that $A$ is an LFD universally going-down domain and $B$ is algebraic over $A$, then the inclusion map $A[X_1,\cdots ,X_n]\hookrightarrow B[X_1,\cdots ,X_n]$ satisfies GB for each $n\ge 0$. However, for any nonzero ring...

A ring extension $R\subseteq S$ is said to be strongly affine if each $R$-subalgebra of $S$ is a finite-type $R$-algebra. In this paper, several characterizations of strongly affine extensions are given. For instance, we establish that if $R$ is a quasi-local ring of finite dimension, then $R\subseteq S$ is integrally closed and strongly affine if and only if $R\subseteq S$ is a Prüfer extension (i.e. $(R,S)$ is a normal pair). As a consequence, the equivalence of strongly affine extensions, quasi-Prüfer extensions and INC-pairs is shown. Let $G$ be...

Let $R$ be a commutative ring with an identity different from zero and $n$ be a positive integer. Anderson and Badawi, in their paper on $n$-absorbing ideals, define a proper ideal $I$ of a commutative ring $R$ to be an $n$-absorbing ideal of $R$, if whenever $x_1\cdots x_{n+1}\in I$ for $x_1,...,x_{n+1}\in R$, then there are $n$ of the $x_i$’s whose product is in $I$ and conjecture that ${\omega }_{R\left[X\right]}\left(I\left[X\right]\right)={\omega }_R\left(I\right)$ for any ideal $I$ of an arbitrary ring $R$, where ${\omega }_R\left(I\right)=min\{n:I\text{is}\text{an}n\text{-absorbing}\text{ideal}\text{of}R\}$. In the present paper, we use content formula techniques to prove that their conjecture is true, if one of the following conditions...

Dato un insieme $X$ di $s$ punti nello spazio proiettivo, si costruisce un esplicito ideale canonico $\mathcal{I}$ nel suo anello di coordinate $R$. Si descrivono le componenti omogenee di $\mathcal{I}$ e la struttura della mappa di moltiplicazione $R_{\sigma }\otimes {\mathcal{I}}_{\sigma +1}\to {\mathcal{I}}_{2\sigma +1}$, dove $\sigma =max\left\{i\mid H_X\left(i\right)<s\right\}$. Tra le applicazioni ci sono varie caratterizzazioni di insiemi di punti coomologicamente uniformi, disuguaglianze nelle loro funzioni di Hilbert, il calcolo del primo modulo delle sizigie di $\mathcal{I}$ in casi particolari, una generalizzazione della «trasformata di Gale» a trasformate...