Related papers: Nil-Essential Ideals

In this paper, using N-structure, the notion of an N-ideal in a BE-algebra is introduced. Conditions for an N-structure to be an N-ideal are provided. To obtain a more general form of an N-ideal, a point N-structure which is (k…

We establish the primary decomposition and uniqueness of primary decomposition for k-ideals in commutative Noetherian semirings.

In this paper we extend the characterisation of kernels in semirings as subtractive ideals to general algebras. We then analyse the counterparts of ``subtractive'' and ``ideal'' in several different algebraic settings.

We survey theory developed over the past 10 years of semirings which need not be additively cancellative. The main feature is a specified ``null ideal'' $\mcA_0$ of a semiring $\mcA,$ taking the place of a zero element, which permits…

Let $R$ be a commutative ring with $ 1 \neq 0$. We recall that a proper ideal $I$ of $R$ is called a semiprimary ideal of $R$ if whenever $a,b\in R$ and $ab \in I$, then $a\in \sqrt{I}$ or $b\in \sqrt{I}$. We say $I$ is a {\it weakly…

We characterise, in pointed regular categories, the ideals as the zero-classes of surjective relations. Moreover, we study a variation of the "Smith is Huq" condition: two surjective left split relations commute as soon as their…

Given a unital associative ring S and a subring R, we say that S is an ideal (or Dorroh) extension of R if for some ideal I of S, S = R + I, where the sum is direct. In this note we investigate the ideal structure of an arbitrary ideal…

Let $(R, \mathfrak{m})$ be a regular local ring of characteristic $p > 0$. Among all proper ideals $\mathfrak{a}\subseteq R$ with a fixed order of vanishing $\text{ord}_{\mathfrak{m}}(\mathfrak{a})$, we classify the ideals for which the…

We investigate ideal-semisimple and congruence-semisimple semirings. We give several new characterizations of such semirings using e-projective and e-injective semimodules. We extend several characterizations of semisimple rings to (not…

We expand the notion of core to $cl$-core for Nakayama closures $cl$. In the characteristic $p>0$ setting, when $cl$ is the tight closure, denoted by *, we give some examples of ideals when the core and the *-core differ. We note that…

We consider and study those rings in which each nil-clean or clean element is uniquely nil-clean. We establish that, for abelian rings, these rings have a satisfactory description and even it is shown that the classes of abelian rings and…

Among the several types of closures of an ideal $I$ that have been defined and studied in the past decades, the integral closure $\bar{I}$ has a central place being one of the earliest and most relevant. Despite this role, it is often a…

This paper introduces and studies nil-reversible rings wherein we call a ring R nil-reversible if the left and right annihilators of every nilpotent element of R are equal. Reversible rings (and hence reduced rings) form a proper subclass…

Let G be a finite p-group which does not contain a rank two elementary abelian p-group as a direct factor. Then the ideal of essential classes in the mod-p cohomology ring of G is a Cohen-Macaulay module whose Krull dimension is the p-rank…

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…

Let $S$ be a commutative ring with identity and $R$ a unitary subring of $S$. An ideal $I$ of $S$ is called an $R$-conductor ideal of $S$ if $I=\{x\in S\mid xS\subseteq V\}$ for some intermediate ring $V$ of $R$ and $S$. In this note we…

A semiring is uniserial if its ideals are totally ordered by inclusion. First, we show that a semiring $S$ is uniserial if and only if the matrix semiring $M_n(S)$ is uniserial. As a generalization of valuation semirings, we also…

We introduce and study a special class of ideals, called tropical ideals, in the semiring of tropical polynomials, with the goal of developing a useful and solid algebraic foundation for tropical geometry. The class of tropical ideals…

We study integrality over rings (all commutative in this paper) and over ideal semifiltrations (a generalization of integrality over ideals). We begin by reproving classical results, such as a version of the "faithful module" criterion for…

We develop a general ring theory in the o-minimal setting culminating in a description of all the definable rings in an arbitrary o-minimal structure. We show that every definably connected ring with non-trivial multiplication defines an…