相关论文: Star-reductions of ideals and Prufer v-multiplicat…

Let $R$ be a commutative integral domain and let $\star$ be a semistar operation of finite type on $R$, and $I$ be a quasi-$\star$-ideal of $R$. We show that, if every minimal prime ideal of $I$ is the radical of a $\star$-finite ideal,…

In this paper, we consider five possible extensions of the Pr\"ufer domain notion to the case of commutative rings with zero-divisors. We investigate the transfer of these Pr\"ufer-like properties between a ring $R$ and $R\bowtie I$; his…

Let $R$ be a commutative ring with unity $(1\not=0)$ and let $\mathfrak{J}(R)$ be the set of all ideals of $R$. Let $\phi:\mathfrak{J}(R)\rightarrow\mathfrak{J}(R)\cup\{\emptyset\}$ be a reduction function of ideals of $R$ and let…

Let $R$ be a commutative ring with identity. In this note, we study the property: If $ I \subsetneqq J$ are ideals in $R$, then $ I^n \subsetneqq J^n$ for all $ n\geq 1$. We define the notion of a big ideal (Definition 1.2). It is noted…

Let $\ast$ be a star operation on an integral domain $D$. Let $\f(D)$ be the set of all nonzero finitely generated fractional ideals of $D$. Call $D$ a $\ast$--Pr\"ufer (respectively, $(\ast, v)$--Pr\"ufer) domain if $(FF^{-1})^{\ast}=D$…

Let $R=\bigoplus_{\alpha\in\Gamma}R_{\alpha}$ be a graded integral domain and $\star$ be a semistar operation on $R$. For $a\in R$, denote by $C(a)$ the ideal of $R$ generated by homogeneous components of $a$ and…

We show that in certain Pr\"ufer domains, each nonzero ideal $I$ can be factored as $I=I^v \Pi$, where $I^v$ is the divisorial closure of $I$ and $\Pi$ is a product of maximal ideals. This is always possible when the Pr\"ufer domain is…

Let $R$ be a commutative ring with identity. The structure theorem says that $R$ is a PIR (resp., UFR, general ZPI-ring, $\pi$-ring) if and only if $R$ is a finite direct product of PIDs (resp., UFDs, Dedekind domains, $\pi$-domains) and…

Let $\mathcal{I}(R)$ be the set of all ideals of a ring $R$, $\delta$ be an expansion function of $\mathcal{I}(R)$. In this paper, the $\delta$-$J$-ideal of a commutative ring is defined, that is, if $a, b\in R$ and $ab\in I\in…

Let $R$ be a commutative ring with identity. An ideal $I$ of $R$ is said to be a big ideal (resp. an upper big ideal) if whenever $J\subsetneqq I$ (resp. $I\subsetneqq J$), $J^{n}\subsetneqq I^{n}$ (resp. $I^{n}\subsetneqq J^{n}$) for every…

Let R be a commutative ring with identity and M be an R-module. A proper ideal I of R is said to be a $z^\circ$-ideal if for each $a \in I$ the intersection of all minimal prime ideals containing a is contained in I. The purpose of this…

In this paper, we introduce a new generalization of weakly prime ideals called $I$-prime. Suppose $R$ is a commutative ring with identity and $I$ a fixed ideal of $R$. A proper ideal $P$ of $R$ is $I$-prime if for $a, b \in R$ with $ab \in…

For an integral domain D and a star operation * on D, we study the following condition: whenever I>AB with I, A, B nonzero ideals, there exist nonzero ideals H and J such that I*=(HJ)*, H*>A and J*>B.

Let $R$ be a commutative ring, we say that $\mathcal{A}\subseteq Spec(R)$ has prime avoidance property, if $I\subseteq \bigcup_{P\in\mathcal{A}}P$ for an ideal $I$ of $R$, then there exists $P\in\mathcal{A}$ such that $I\subseteq P$. We…

This paper investigates ideal-theoretic as well as homological extensions of the Prufer domain concept to commutative rings with zero divisors in an amalgamated duplication of a ring along an ideal. The new results both compare and contrast…

Let $R$ be a commutative ring with nonzero identity. Let $\mathcal{I}(R)$ be the set of all ideals of $R$ and let $\delta : \mathcal{I}(R)\longrightarrow \mathcal{I}(R)$ be a function. Then $\delta$ is called an expansion function of ideals…

We study stable semistar operations defined over a Pr\"ufer domain, showing that, if every ideal of a Pr\"ufer domain $R$ has only finitely many minimal primes, every such closure can be described through semistar operations defined on…

Let $R$ be a commutative ring with identity. The ring $R\times R$ can be viewed as an extension of $R$ via the diagonal map $\Delta: R \hookrightarrow R\times R$, given by $\Delta(r) = (r, r)$ for all $r\in R$. It is shown that, for any $a,…

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…

A commutative ring R has finite rank r, if each ideal of R is generated at most by r elements. A commutative ring R has the r-generator property, if each finitely generated ideal of R can be generated by r elements. Such rings are closely…