Related papers: Connections between Representation-Finite and K\"o…

Let $R$ be a ring. In this paper, we study the characterization of cosilting modules and establish a relation between cosilting modules and cotilting objects in a Grothendieck category. We proved that each cosilting right $R$-module $T$ can…

Suppose X is a projective toric scheme defined over a commutative ring R equipped with an ample line bundle L. We prove that its K-theory has k+1 direct summands K(R) where k is minimal among non-negative integers such that the twisted line…

Let $K[HK_{\Theta}]$ denote the Hecke-Kiselman algebra of a finite oriented graph $\Theta$ over an algebraically closed field $K$. All irreducible representations, and the corresponding maximal ideals of $K[HK_{\Theta}]$, are characterized…

We consider rings whose one-sided ideals are close to automorphism-invariant modules. We study rings in which every (finitely generated) right ideal is automorphism invariant and rings in which every right ideal is a finite direct sum of…

Given a commutative ring $R$ and finitely generated ideal $I$, one can consider the classes of $I$-adically complete, $L_0^I$-complete and derived $I$-complete complexes. Under a mild assumption on the ideal $I$ called weak pro-regularity,…

We extend the notion of a purely infinite simple C*-algebra to the context of unital rings, and we study its basic properties, specially those related to K-Theory. For instance, if $R$ is a purely infinite simple ring, then $K_0(R)^+=…

If R is a commutative ring, we prove that every finitely generated module has a pure-composition series with indecomposable factors and any two such series are isomorphic if and only if R is a Bezout ring and a CF-ring.

We prove that the tangent complex of K-theory, in terms of (abelian) deformation problems over a characteristic 0 field k, is cyclic homology (over k). This equivalence is compatible with the $\lambda$-operations. In particular, the…

Let $R$ be a commutative $k-$algebra over a field $k$. Assume $R$ is a noetherian, infinite, integral domain. The group of $k-$automorphisms of $R$,i.e.$Aut_k(R)$ acts in a natural way on $(R-k)$.In the first part of this article, we study…

Let k be a commutative algebra with the field of the rational numbers included in k and let (E,p,i) be a cleft extension of A. We obtain a new mixed complex, simpler than the canonical one, giving the Hochschild and cyclic homologies of E…

Let $k$ be a field of characteristic zero and $B$ a commutative integral domain that is also a finitely generated $k$-algebra. It is well known that if $k$ is algebraically closed and the "Field Makar-Limanov" invariant FML$(B)$ is equal to…

The copointed liftings of the Fomin-Kirillov algebra $\mathcal{FK}_3$ over the algebra of functions on the symmetric group $\mathbb{S}_3$ were classified by Andruskiewitsch and the author. We demonstrate here that those associated to a…

Starting from the notion of totally reflexive modules, we survey the theory of Gorenstein homological dimensions for modules over commutative rings. The account includes the theory's connections with relative homological algebra and with…

Let G be a real, connected, noncompact, semisimple Lie group, let K be a maximal compact subgroup of G, and let g=k+p be the corresponding Cartan decomposition of the complexified Lie algebra of G. Sequences of strongly orthogonal…

A cover by left ideals of an associative (not necessarily commutative or unital) ring $R$ is a collection of proper left ideals whose set-theoretic union equals $R$. If such a cover exists, then $\eta_\ell(R)$ is the cardinality of a…

In the present paper the cyclic homology functor from the category of $A_\infty$-algebras over any commutative unital ring $K$ to the category of graded $K$-modules is constructed. Further, it is showed that this functor sends homotopy…

The (co)completeness problem for the (projectively) stable module category of an associative ring is studied. (Normal) monomorphisms and (normal) epimorphisms in such a category are characterized. As an application, we give a criterion for…

We associate to any ring $R$ with identity a partially ordered set Hom$(R)$, whose elements are all pairs $(\mathfrak a,M)$, where $\mathfrak a=\ker\varphi$ and $M=\varphi^{-1}(U(S))$ for some ring morphism $\varphi$ of $R$ into an…

Let $R=C[[t]]$ be the ring of power series over an algebraically closed field $C$ of characteristic zero. We show that each connection on a finite flat $R((x))$-module is the sum of a regular singular connection and a diagonalizable…

Fix a ring $ R $ and look at the class of left $ R $-modules and naturally, we restrict ourselves to the case of rings such that this class is not too similar to the case $ R $ is a field. We shall solve Kaplansky test problems, all three…