Related papers: Is an irng singly generated as an ideal?
A ring $R$ is a UU ring if every unit is unipotent, or equivalently if every unit is a sum of a nilpotent and an idempotent that commute. These rings have been investigated in C\u{a}lug\u{a}reanu \cite{C} and in Danchev and Lam \cite{DL}.…
Absolute integral closures of general commutative unital rings are explored. All rings admit absolute integral closures, but in general they are not unique. Among the reduced rings with finitely many minimal prime ideals, finite products of…
Can there be a structure space-type theory for an arbitrary class of ideals of a ring? The ideal spaces introduced in this paper allows such a study and our theory includes (but not restricted to) prime, maximal, minimal prime, strongly…
We prove that a commutative parasemifield S is additively idempotent provided that it is finitely generated as a semiring. Consequently, every proper commutative semifield T that is finitely generated as a semiring is either additively…
We study rings over which an analogue of the Weierstrass preparation theorem holds for power series. We show that a commutative ring $R$ admits a factorization of every power series in $R[[x]]$ as the product of a polynomial and a unit if…
Completely prime right ideals are introduced as a one-sided generalization of the concept of a prime ideal in a commutative ring. Some of their basic properties are investigated, pointing out both similarities and differences between these…
In this paper we introduce and study a graph on the set of ideals of a commutative ring $R$. The vertices of this graph are non-trivial ideals of $R$ and two distinct ideals $I$ and $J$ are adjacent if and only $IJ=I\cap J$. We obtain some…
Let R be a commutative noetherian ring. Lindo and Pande have recently posed the question asking when every ideal of R is isomorphic to some trace ideal of R. This paper studies this question and gives several answers. In particular, a…
For $R_1,R_2,R_3,\dots$ a family of non isomorphic rings (or algebras) having each only 2 idempotents ($1$ and $0$), we classify up to isomorphism the rings (or algebras) obtained by taking products of powers of the different $R_i$. We show…
A group has normal rank (or weight) greater than one if no single element normally generates the group. The Wiegold problem from 1976 asks about the existence of a finitely generated perfect group of normal rank greater than one. We show…
We give an explicit and character-free construction of a complete set of orthogonal primitive idempotents of a rational group algebra of a finite nilpotent group and a full description of the Wedderburn decomposition of such algebras. An…
In this paper, we inspect a relatively unexplored notion of finite generation in semirings, namely semirings in which all congruences are finitely generated. Such semirings are dubbed Congruence Noetherian. After developing sufficient…
Let $R$ be a commutative ring with identity. A unit $u$ of $R$ is called exceptional if $1-u$ is also a unit. When $R$ is a finite commutative ring, we determine the additive and multiplicative structures of its exceptional units; and then…
An element $x \in R$ is considered (strongly) nil-clean if it can be expressed as the sum of an idempotent $e \in R$ and a nilpotent $b \in R$ (where $eb = be$). If for any $x \in R$, there exists a unit $u \in R$ such that $ux$ is…
We give a positive answer to the Huneke-Wiegand Conjecture for monomial ideals over free numerical semigroup rings, and for two generated monomial ideals over complete intersection numerical semigroup rings.
A longstanding open question is whether every strongly clean ring (ring in which every element is strongly clean, i.e., is the sum of an idempotent and a unit which commute with each other) is Dedekind-finite (has the property that every…
The three famous problems concerning units, zero-divisors and idempotents in group rings of torsion-free groups, commonly attributed to I. Kaplansky, have been around for more than 60 years and still remain open in characteristic zero. In…
We consider in-depth and characterize in certain aspects those rings whose non-units are strongly nil-clean in the sense that they are a sum of commuting nilpotent and idempotent. In addition, we examine those rings in which the non-units…
A unital ring is called clean (resp. strongly clean) if every element can be written as the sum of an invertible element and an idempotent (resp. an invertible element and an idempotent that commutes). T.Y. Lam proposed a question: which…
We compute the finite generation ideal for Daigle and Freudenburg's counterexample to Hilbert's fourteenth problem. This ideal helps to understand how far the ring of invariants is from being finitely generated. Our calculations show that…