We show in an additive inverse regular semiring $(S,+,·)$ with $E^{•}\left(S\right)$ as the set of all multiplicative idempotents and $E^{+}\left(S\right)$ as the set of all additive idempotents, the following conditions are equivalent: (i) For all $e,f\in E^{•}\left(S\right)$, $ef\in E^{+}\left(S\right)$ implies $fe\in E^{+}\left(S\right)$. (ii) $(S,·)$ is orthodox. (iii) $(S,·)$ is a semilattice of groups. This result generalizes the corresponding result of regular ring.

We provide a characterization of artin algebras without chains of nonzero homomorphisms between indecomposable finitely generated modules starting with an injective module and ending with a projective module.

We discuss the existence of tilting modules which are direct limits of finitely generated tilting modules over tilted algebras.

For a locally convex *-algebra A equipped with a fixed continuous *-character ε (which is roughly speaking a generalized F*-algebra), we define a cohomological property, called property (FH), which is similar to character amenability. Let $C_c\left(G\right)$ be the space of continuous functions with compact support on a second countable locally compact group G equipped with the convolution *-algebra structure and a certain inductive topology. We show that $(C_c\left(G\right),{\epsilon }_G)$ has property (FH) if and only if G has property (T). On...

A trivializability principle for local rings is described which leads to a form of weak algorithm for local semifirs with a finitely generated maximal ideal whose powers meet in zero.

Given a smooth proper dg algebra $A$, a perfect dg $A$-module $M$ and an endomorphism $f$ of $M$, we define the Hochschild class of the pair $(M,f)$ with values in the Hochschild homology of the algebra $A$. Our main result is a Riemann-Roch type formula involving the convolution of two such Hochschild classes.

Let $D$ be a holomorphic differential operator acting on sections of a holomorphic vector bundle on an $n$-dimensional compact complex manifold. We prove a formula, conjectured by Feigin and Shoikhet, giving the Lefschetz number of $D$ as the integral over the manifold of a differential form. The class of this differential form is obtained via formal differential geometry from the canonical generator of the Hochschild cohomology $H{H}^{2n}({𝒟}_n,{𝒟}_n^{*})$ of the algebra of differential operators on a formal neighbourhood of a...

This text gives a short overview of the recent works on Gorenstein global dimension of rings.

In this paper, we introduce a subclass of strongly clean rings. Let $R$ be a ring with identity, $J$ be the Jacobson radical of $R$, and let $J^{\#}$ denote the set of all elements of $R$ which are nilpotent in $R/J$. An element $a\in R$ is called very $J^{\#}$-clean provided that there exists an idempotent $e\in R$ such that $ae=ea$ and $a-e$ or $a+e$ is an element of $J^{\#}$. A ring $R$ is said to be very $J^{\#}$-clean in case every element in $R$ is very $J^{\#}$-clean. We prove that every very $J^{\#}$-clean ring is strongly $\pi$-rad clean and has stable range one. It is shown...

This article presents a brief survey of the work done on rings generated by their units.

This paper discusses a variant theory for the Gorenstein flat dimension. Actually, since it is not yet known whether the category (R) of Gorenstein flat modules over a ring R is projectively resolving or not, it appears legitimate to seek alternate ways of measuring the Gorenstein flat dimension of modules which coincide with the usual one in the case where (R) is projectively resolving, on the one hand, and present nice behavior for an arbitrary ring R, on the other. In this paper, we introduce...