Related papers: Finitely generated and not finitely generated ring…

Let $R$ be a commutative ring with identity and $S$ a multiplicative subset of $R$. First, we introduce and study the $S$-projective dimensions and $S$-injective dimensions of $R$-modules, and then explore the $S$-global dimension…

This work is a review of results about centrally essential rings and semirings. A ring (resp., semiring) is said to be centrally essential if it is either commutative or satisfy the property that for any non-central element $a$, there exist…

In this paper we study (non-commutative) rings $R$ over which every finitely generated left module is a direct sum of cyclic modules (called left FGC-rings). The commutative case was a well-known problem studied and solved in 1970s by…

In this note we consider a notion of relative Frobenius pairs of commutative rings $S/R$. To such a pair, we associate an $\mathbb{N}$-graded $R$-algebra $\Pi_R(S)$ which has a simple description and coincides with the preprojective algebra…

We study varieties with a finitely generated Cox ring. In a first part, we generalize a combinatorial approach developed in earlier work for varieties with a torsion free divisor class group to the case of torsion. Then we turn to…

A centrally essential ring is a ring which is an essential extension of its center (we consider the ring as a module over its center). We give several examples of noncommutative centrally essential rings and describe some properties of…

We develop a technique to construct finitely injective modules which are non trivial, in the sense that they are not direct sums of injective modules. As a consequence, we prove that a ring $R$ is left noetherian if and only if each…

We classify all non-degenerate skew-hermitian forms defined over certain local rings, not necessarily commutative, and study some of the fundamental properties of the associated unitary groups, including their orders when the ring in…

Let $R$ be a finite ring. The commuting probability of $R$ is the probability that any two randomly chosen elements of $R$ commute. In this paper, we obtain some bounds for commuting probability of $R$.

Let $R$ be a commutative Noetherian ring, $\fa$ an ideal of $R$ and $M$ a finitely generated $R$--module. Let $t$ be a non-negative integer such that $\H^i_\fa(M)$ is $\fa$--cofinite for all $i<t$. It is well--known that…

Hard to summarize concisely; here are the high points. The first two statements below are ring-theoretic; in these R is a nontrivial ring, R^\omega, and \bigoplus_\omega R are the direct product, respectively direct sum, of countably many…

We construct a commutative version of the group ring and show that it allows one to translate questions about the normal generation of groups into questions about the generation of ideals in commutative rings. We demonstrate this with an…

In this paper we study the probability that the commutator of a randomly chosen pair of elements, one from a subring of a finite ring and other from the ring itself equals to a given element of the ring.

Let $R$ be a commutative ring with identity and let $M$ be an $R$-module which is generated by $\mu$ elements but not fewer. We denote by $\operatorname{SL}_n(R)$ the group of the $n \times n$ matrices over $R$ with determinant $1$. We…

We define the notions of finite-state and functionally recursive matrices and their growth. We also introduce two rings generated by functionally recursive matrices. The first is isomorphic to the 2-generated free ring. The second is a…

For a projective curve $C$ and the canonical divisor $K_C$ on $C$, it is classically known that the canonical ring $R(C) = \oplus_{m=0}^\infty H^0(C, m K_C)$ is finitely generated in degree at most three. In this article, we study whether…

Let $R$ be a ring and let $A$ be a finite projective $R$-algebra of rank $n$. Manjul Bhargava and Matthew Satriano have recently constructed an $R$-algebra $G(A/R)$, the Galois closure of $A/R$. Many natural questions were asked at the end…

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…

The purpose of this work is to extend the study of the commutative rings whose lattice of ideals can be a structure of BL-algebra as carry out by Heubo et al in 2018, to non commutative rings appointed in the work as pseudo BL-rings. We…

We present new characterizations of the rings in which every element is the sum of two idempotents and a nilpotent that commute, and the rings in which every element is the sum of two tripotents and a nilpotent that commute. We prove that…