Narrow Results

A proper ideal $I$ in a ring $R$ is called strongly irreducible (briefly, SI-ideal), if $J\cap K\subseteq I$ then $J\subseteq I$ or $K\subseteq I$, where $J$, $K$ are ideals of $R$. We show that whenever in a local Noetherian duo domain, a power of its maximal ideal, $M$ say, is principal, then so is $M$. It is also shown that whenever $I$ is an SI-ideal in a Noetherian duo ring $R$ where $\sqrt{I}=P$ and $I\ne P$ with $\mathrm{\text{ht}}(P)>0$, then $R_P$ is a domain. Finally in a Noetherian duo ring $R$, non-prime ideals with positive heights, which are SI-ideal, are characterized.

Let $R$ be a commutative ring with nonzero identity. In this paper, we introduce the concept of 1-absorbing primary ideals in commutative rings. A proper ideal $I$ of $R$ is called a $1$-absorbing primary ideal of $R$ if whenever nonunit elements $a,b,c\in R$ and $abc\in I$, then $ab\in I$ or $c\in \sqrt{I}.$ Some properties of 1-absorbing primary ideals are investigated. For example, we show that if $R$ admits a 1-absorbing primary ideal that is not a primary ideal, then $R$ is a quasilocal ring. We give an example of a 1-absorbing primary ideal of $R$ that is not a primary ideal of $R$. We show that if $R$ is a Noetherian domain, then $R$ is a Dedekind domain if and only if every nonzero proper 1-absorbing primary ideal of $R$ is of the form $P^n$ for some nonzero prime ideal $P$ of $R$ and a positive integer $n\ge 1$. We show that a proper ideal $I$ of $R$ is a 1-absorbing primary ideal of $R$ if and only if whenever $I_1{I}_2{I}_3\subseteq I$ for some proper ideals $I_1,I_2,I_3$ of $R$, then $I_1{I}_2\subseteq I$ or $I_3\subseteq \sqrt{I}.$

Let $R$ be a commutative ring with identity. In this paper, we introduce the concept of $1$-absorbing prime ideals which is a generalization of prime ideals. A proper ideal $I$ of $R$ is called $1$-absorbing prime if for all nonunit elements $a,b,c\in R$ such that $abc\in I$, then either $ab\in I$ or $c\in I$. Some properties of $1$-absorbing prime are studied. For instance, it is shown that if $R$ admits a $1$-absorbing prime ideal that is not a prime ideal, then $R$ is a quasi–local ring. Among other things, it is proved that a proper ideal $I$ of $R$ is $1$-absorbing prime if and only if the inclusion $I_1{I}_2{I}_3\subseteq I$ for some proper ideals $I_1,I_2,I_3$ of $R$ implies that $I_1{I}_2\subseteq I$ or $I_3\subseteq I$. Also, $1$-absorbing prime ideals of PIDs, valuation domains, Prufer domains and idealization of a modules are characterized. Finally, an analogous to the Prime Avoidance Theorem and some applications of this theorem are given.

Let $O$ be an order in a quadratic number field $K$ with ring of integers $D$, such that the conductor $𝔉=fD$ is a prime ideal of $O$, where $f\in \mathbb{Z}$ is a prime. We give a complete description of the $𝔉$-primary ideals of $O$. They form a lattice with a particular structure by layers; the first layer, which is the core of the lattice, consists of those $𝔉$-primary ideals not contained in ${𝔉}^2$. We get three different cases, according to whether the prime number $f$ is split, inert or ramified in $D$.

Let $R$ be a commutative ring with unity $1\ne 0$ and let $\delta$ be an ideal expansion. In the first part of this paper, we extend the concept of $\delta$-primary ideals to $\varphi$-$\delta$-primary ideals, where $\varphi$ is an ideal reduction and $\delta$ is an ideal expansion. We introduce some of the ideal expansion $\delta$ and define $\varphi$-$\delta$-primary ideals, where $\varphi$ is an ideal reduction. Also, we investigate ideal expansions satisfying some additional conditions and prove more properties of the generalized $\varphi$-$\delta$-primary ideals with respect to such an ideal expansion $\delta$. In the second part of this paper we investigate 2-absorbing $\delta$-primary ideals which unify 2-absorbing ideals and 2-absorbing primary ideals, where $\delta$ is an ideal expansion. A number of results in the two parts are given.