Isomorphism of modules under ground ring extensions

Let D be the ring of algebraic integers in a number field and Λ a finite D-algebra. It is shown that if M and N are finitely generated Λ-modules, then M and N are locally isomorphic if and only if M and N become isomorphic over some integral extension of D.     

n-Star modules over ring extensions

We give conditions under which an n-star module extends to an n-star module, or an n-tilting module, over a ring extension R of A. In case that R is a split extension of A by Q, we obtain that is a 1-tilting module (respectively, a 1-star module) if and only if is a 1-tilting module (respectively, a 1-star module) and generates both and (respectively, generates ), where is an injective cogenerator in the category of all left A-modules. These extend results in [I. Assem, N. Marmaridis, Tilting modules over split-by-nilpotent extensions, Comm. Algebra 26 (1998) 1547–1555; K.R. Fuller, *-Modules over ring extensions, Comm. Algebra 25 (1997) 2839–2860] by removing the restrictions on R and Q.     

A generalization of silting modules and Tor-tilting modules

We introduce the concept of weak silting modules, which is a generalization of both silting modules and Tor-tilting modules. It is shown that W is a weak silting module if and only if its character module W⁺ is cosilting. Some properties of weak silting modules are given.     

Cancellation of projective modules over polynomial extensions over a two-dimensional ring

Let R be a commutative Noetherian ring of dimension two with $1/2\in R$ and let $A=R[X_1,?,X_n]$. Let P be a projective A-module of rank 2. In this article, we prove that P is cancellative if ${\wedge }^2(P)\oplus A$ is cancellative.     

Distributive extensions of modules

Let X be a submodule of a module M. The extension is said to be distributive if X ∩ (Y + Z) = X ∩ Y + X ∩ Z for any two submodules Y and Z of M. We study distributive extensions of modules over not necessarily commutative rings. In particular, it is proved that the following three conditions are equivalent: (1) is a distributive extension; (2) for any submodule Y of the module M, no simple subfactor of the module X/(X∩Y ) is isomorphic to any simple subfactor of Y/(X∩Y) (3) for any two elements x ∈ X and m ∈ M, there does not exist a simple factor module of the cyclic module xA/(X ∩ mA) that is isomorphic to a simple factor module of the cyclic module mA/(X ∩ mA). __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 12, No. 3, pp. 141–150, 2006.     

Gorenstein projective modules and Frobenius extensions

We prove that for a Frobenius extension, if a module over the extension ring is Gorenstein projective,then its underlying module over the base ring is Gorenstein projective; the converse holds if the frobenius extension is either left-Gorenstein or separable(e.g., the integral group ring extension ZZG).Moreover, for the Frobenius extension RA = R[x]/(x~2), we show that: a graded A-module is Gorenstein projective in GrMod(A), if and only if its ungraded A-module is Gorenstein projective, if and only if its underlying R-module is Gorenstein projective. It immediately follows that an R-complex is Gorenstein projective if and only if all its items are Gorenstein projective R-modules.     

On extensions of rational modules

Given a topological algebra A, we investigate when the categories of all rational A-modules and of finite-dimensional rational modules are closed under extensions inside the category of A-modules. We give a complete characterization of these two properties, in terms of a topological and a homological condition, for complete algebras. We also give connections to other important notions in coalgebra theory such as coreflexive coalgebras. In particular, we are able to generalize many previously known partial results and answer some questions in this direction, and obtain large classes of coalgebras for which rational modules are closed under extensions as well as various examples where this is not true.     

Tilting modules over split-by-nilpotent extensions

Let a finite dimensional algebra R be a split extension of an algebra A by a nilpotent bimodule Q. We give necessary and sufficient conditions for a (partial) tilting module T_A to be such that T?_A R_R is a (partial) tilting module. If this is not the case, but Q_A is generated by the tilting module T_A , then there exists a quotient [Rbar] of R such that T?_A [Rbar]_[Rbar] is a tilting module.     

Minimal numerical-radius extensions of operators

In this paper we characterize minimal numerical-radius extensions of operators from finite-dimensional subspaces and compare them with minimal operator-norm extensions. We note that in the cases , and in the case of self-adjoint extensions in , the two extensions and their norms are equal.

We also show that, in the case of , , and more generally in the case of the dual space being strictly convex, if the minimal projections with respect to the operator norm and with respect to the numerical radius have equal norms, then the operator norm is . An analogous result is also true for an arbitrary extension. Finally, we provide an example of a projection from onto a two-dimensional subspace which is minimal with respect to norm but not with respect to the numerical radius for , and we determine the minimal numerical-radius projection in this same situation.

     

Integral ring extensions and lo-pairs

We give an example of an LO-pair (R,T) with dim R=dimT = n, so that all maximal ideals of R have the same height n, but with T not integral over R. This answers an open question of David E. Dobbs.     

Armendariz modules over skew PBW extensions

The aim of this article is to develop the theory of skew Armendariz and quasi-Armendariz modules over skew PBW extensions. We generalize the results of several works in the literature concerning these topics for the context of Ore extensions to another non-commutative rings which can not be expressed as iterated Ore extensions. As a consequence of our treatment, we extend and unify different results about the Armendariz, Baer, p.p., and p.q.-Baer properties for Ore extensions and skew PBW extensions.