Related papers: Idempotent pairs and PRINC domains

In this note we answer the question raised by Han et al. in J. Korean Math. Soc (2014) whether an idempotent isomorphic to a semicentral idempotent is itself semicentral. We show that rings with this property are precisely the…

Let $R$ be a Dedekind ring, $K$ its quotient field, and $L=K(\alpha)$ a finite field extension of $K$ defined by a monic irreducible polynomial $f(x)\in R[x]$. We give an easy version of Dedekind's criterion which computationally improves…

A linear relation $E$ acting on a Hilbert space is idempotent if $E^2=E.$ A triplet of subspaces is needed to characterize a given idempotent: $(\mathrm{ran} \, E, \mathrm{ran}(I-E), \mathrm{dom}\, E),$ or equivalently, $(\mathrm{ker}(I-E),…

An integral domain is said to have the IDF property when every non-zero element of it has only a finite number of non-associate irreducible divisors. A counterexample has already been found showing that IDF property does not necessarily…

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…

Let R* be an ideal-adic completion of a Noetherian integral domain R and let L be a subfield of the total quotient ring of R* such that L contains R. Let A denote the intersection of L with R*. The integral domain A sometimes inherits nice…

An integral domain $D,$ with quotient field $K,$ is a $v$-domain if for each nonzero finitely generated ideal $A$ of $D$ we have $(AA^{-1})^{-1}=D.$ It is well known that if $D$ is a $v$-domain$,$ then some quotient ring $D_{S}$ of $D$ may…

In 2020, Cossu and Zanardo raised a conjecture on the idempotent factorization on singular matrices in the form $\begin{pmatrix} p&z\\ \bar{z}&\sfrac{\lVert z\rVert}{p} \end{pmatrix},$ where $p$ is a prime integer which is irreducible but…

For a principal ideal domain $A$, the Latimer--MacDuffee correspondence sets up a bijection between the similarity classes of matrices in $\operatorname{M}_{n}(A)$ with irreducible characteristic polynomial $f(x)$ and the ideal classes of…

In this article we investigate when a homogeneous ideal in a graded ring is normal, that is, when all positive powers of the ideal are integrally closed. We are particularly interested in homogeneous ideals in an N-graded ring generated by…

We study a ring containing a complete set of orthogonal idempotents as a generalized matrix ring via its Peirce decomposition. We focus on the case where some of the underlying bimodule homomorphisms are zero. Upper and lower triangular…

In this paper, new algebraic and topological results on purely-prime ideals of a commutative ring (pure spectrum) are obtained. Especially, Grothendieck type theorem is obtained which states that there is a canonical correspondence between…

An integral domain is atomic if every nonzero nonunit factors into irreducibles. Let $R$ be an integral domain. We say that $R$ is a bounded factorization domain if it is atomic and for every nonzero nonunit $x \in R$, there is a positive…

An integral domain $D$ is a {\em valuation ideal factorization domain} (VIFD) if each nonzero principal ideal of $D$ can be written as a finite product of valuation ideals. Clearly, $\pi$-domains are VIFDs. We study the ring-theoretic…

We study certain (two-sided) nil ideals and nilpotent ideals in a Lie nilpotent ring R. Our results lead us to showing that the prime radical rad(R) of R comprises the nilpotent elements of R, and that if L is a left ideal of R, then…

Idempotents yield much insight in the structure of finite semigroups and semirings. In this article, we obtain some results on (multiplicatively) idempotents of the endomorphism semiring of a finite chain. We prove that the set of all…

An irreducible element of a commutative ring is absolutely irreducible if no power of it has more than one (essentially different) factorization into irreducibles. In the case of the ring $\text{Int}(D)=\{f\in K[x]\mid f(D)\subseteq D\}$,…

A (positive definite and integral) quadratic form is said to be $\textit{prime-universal}$ if it represents all primes. Recently, Doyle and Williams in [2] classified all prime-universal diagonal ternary quadratic forms, and all…

A complete understanding of the structure of all prime ideals of an infinite direct product of commutative rings (e.g. in terms of more specific objects) has remained a challenging problem for decades. In this article, new advances have…

Let $D$ be an integral domain with quotient field $K$ and let $\mathcal{I}% (D) $ be the set of nonzero ideals of $D$. Call, for $I,J\in \mathcal{I}(D)$ , the product $IJ$ of ideals condensed if $IJ=\{ij|i\in I,j\in J\}.$ Call $D$ a…