Related papers: On Gelfand graded commutative rings

In commutative ring theory, there is a theorem of Cohen which states that if in a commutative ring all prime ideals are finitely generated then every ideal is finitely generated. However, it is known that having only maximal ideals finitely…

In the first section of this paper, we introduce the notions of fractional and invertible ideals of semirings and characterize invertible ideals of a semidomain. In section two, we define Pr\"{u}fer semirings and characterize them in terms…

We define the Hochschild complex and cohomology of a ring object in a monoidal category enriched over abelian groups. We interpret the cohomology groups and prove that the cohomology ring is graded-commutative.

We describe a collection of differential graded rings that categorify weight spaces of the positive half of the quantized universal enveloping algebra of the Lie superalgebra gl(m|1).

Let $A$ be a regular ring containing a field $K$ of characteristic zero and let $R = A[X_1,\ldots, X_m]$. Consider $R$ as standard graded with $\deg A = 0$ and $\deg X_i = 1$ for all $i$. Let $G$ be a finite subgroup of $GL_m(A)$. Let $G$…

We study the ring extensions R \subseteq T having the same set of prime ideals provided Nil(R) is a divided prime ideal. Some conditions are given under which no such T exist properly containing R. Using idealization theory, the examples…

We consider the ideal structure of Steinberg algebras over a commutative ring with identity. We focus on Hausdorff groupoids that are strongly effective in the sense that their reductions to closed subspaces of their unit spaces are all…

The current paper is dedicated to the study of the classical $K_1$ groups of graded rings. Let $A$ be a $\Gamma$ graded ring with identity $1$, where the grading $\Gamma$ is an abelian group. We associate a category with suspension to the…

Let $G$ be a group. A ring $R$ is called a graded ring (or $G$-graded ring) if there exist additive subgroups $R_{\alpha }$ of $R$ indexed by the elements $\alpha \in G$ such that $R=\bigoplus_{\alpha \in G}R_{\alpha }$ and $R_{\alpha…

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…

Let $A$ and $B$ be two connected graded algebras finitely generated in degree one. If $A$ is isomorphic to $B$ as ungraded algebras, then they are also isomorphic to each other as graded algebras.

In this article, we define the concept of an $S$-$k$-irreducible ideal and $S$-$k$-maximal ideal in a commutative semiring. We also establish several results concerning $S$-$k$-primary ideals and prove the existence theorem and the…

We study criteria for a ring - or more generally, for a small category - to be Gorenstein and for a module over it to be of finite projective dimension. The goal is to unify the universal coefficient theorems found in the literature and to…

Let R be a commutative ring with identity. In this paper, we introduce the concept 1-absrbing primary ideal of R.

Given a $1$-tilting cotorsion pair over a commutative ring, we characterise the rings over which the $1$-tilting class is an enveloping class. To do so, we consider the faithful finitely generated Gabriel topology $\mathcal{G}$ associated…

We introduce a new class of graded rings extending the class of generalized Weyl algebras. These rings are orders in crossed products of the most general type, and we introduce their basic structure theory. We provide an extensive list of…

We consider ideals in a polynomial ring that are generated by regular sequences of homogeneous polynomials and are stable under the action of the symmetric group permuting the variables. In previous work, we determined the possible…

Let $R$ be a commutative ring. We investigate $R$-modules which can be written as \emph{finite} sums of {\it {second}} $R$-submodules (we call them \emph{second representable}). We provide sufficient conditions for an $R$-module $M$ to be…

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. An ideal $I$ of $R$ is said to be a big ideal (resp. an upper big ideal) if whenever $J\subsetneqq I$ (resp. $I\subsetneqq J$), $J^{n}\subsetneqq I^{n}$ (resp. $I^{n}\subsetneqq J^{n}$) for every…