Related papers: Conch Maximal Subrings

Let $H$ be a connected Hopf algebra acting on an algebra $A$. Working over a base field having characteristic $0$, we show that for a given prime (semi-prime, completely prime) ideal $I$ of $A$, the largest $H$-stable ideal of A contained…

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…

We study the maximal subgroups (also known as group $\mathcal{H}$-classes) of finitely presented special inverse monoids. We show that the maximal subgroups which can arise in such monoids are exactly the recursively presented groups, and…

This article discusses a way for uniquely setting up the valuations for the minimal generators of the maximal ideal of a one dimensional complete reduced and irreducible local algebra over an algebraically closed field, when treated as a…

For $V$ a vector space over a field, or more generally, over a division ring, it is well-known that every $x\in\mathrm{End}(V)$ has an <i>inner inverse</i>, i.e., an element $y\in\mathrm{End}(V)$ satisfying $xyx=x.$ We show here that a…

We show that every Dedekind domain $R$ lying between the polynomial rings $\mathbb Z[X]$ and $\mathbb Q[X]$ with the property that its residue fields of prime characteristic are finite fields is equal to a generalized ring of integer-valued…

In 1969 H\"ochster proved that for every quasi-compact T1-space $X$ we can find a commutative ring $R$ such that $X$ is homeomorphic to the maximal spectrum $\mathrm{Specm}(R)$ of $R$. This result implies the existence of a commutative ring…

For any pseudoconvex Runge domain $\Omega\subset\mathbb{C}^2$ we prove that every closed discrete subset in $\Omega$ is contained in a properly embedded complex curve in $\Omega$ with any prescribed topology (possibly infinite).

We prove that a local domain $R$, essentially of finite type over a field, is regular if and only if for every regular alteration $\pi : X \to Spec R$, we have that $R \pi_* \mathcal{O}_X$ has finite (equivalently zero in characteristic…

Let $R$ be an associative unital algebra over a field $k,$ let $p$ be an element of $R,$ and let $R'=R\langle q\mid pqp= p\rangle.$ We obtain normal forms for elements of $R',$ and for elements of $R'$-modules arising by extension of…

The direct summand conjecture asserts that if R is a regular local ring and S is a module-finite R-algebra containing R, then R is a direct summand of S as an R-module. It was previously known to be true if R contains a field or if dim R is…

We obtain several fundamental results on finite index ideals and additive subgroups of rings as well as on model-theoretic connected components of rings, which concern generating in finitely many steps inside additive groups of rings. Let…

Recognizing when a ring is a complete matrix ring is of significant importance in algebra. It is well-known folklore that a ring $R$ is a complete $n\times n$ matrix ring, so $R\cong M_{n}(S)$ for some ring $S$, if and only if it contains a…

Let $ M (X)$ be the ring of all real measurable functions on a measurable space $(X, \mathscr{A})$. In this article, we show that every ideal of $M(X)$ is a $Z^{\circ}$-ideal. Also, we give several characterizations of maximal ideals of…

Fix any field $K$ of characteristic $p$ such that $[K:K^p]$ is finite. We discuss excellence for Noetherian domains whose fraction field is $K$, showing for example, that $R$ is excellent if and only if the Frobenius map is finite on $R$.…

In this paper we study the modules $M$ every simple subfactors of which is a homomorphic image of $M$ and call them co-Kasch modules. These modules are dual to Kasch modules $M$ every simple subfactors of which can be embedded in $M$. We…

This paper is concerned with existence of big tight closure test elements for a commutative Noetherian ring $R$ of prime characteristic $p$. Let $R^{\circ}$ denote the complement in $R$ of the union of the minimal prime ideals of $R$. A big…

Thirty years ago, Huneke (for local rings) and Lyubeznik (in general) conjectured that for all regular rings $R$, the local cohomology modules $H^i_I(R)$ have finitely many associated prime ideals. We prove substantial new cases of their…

Let $S$ be an integral domain with field of fractions $F$ and let $A$ be an $F$-algebra having an $S$-stable basis. We prove the existence of an $S$-subalgebra $R$ of $A$ lying over $S$ whose localization with respect to $S$ is $A$ (we call…

In this paper, leveraging the recent achievements of researchers, we have revisited the family of ideals of product of commutative rings. We demonstrate that if $ \{ R_\alpha \}_{\alpha \in A} $ is an infinite family of rings, then $ \left|…