Jan Žemlička

Mod-Retractable Rings

A right module M over a ring R is said to be retractable if Hom R (M, N) ≠ 0 for each nonzero submodule N of M. We show that M ⊗ R RG is a retractable RG-module if and only if M R is retractable for every finite group G. The ring R is (finitely) mod-retractable if every (finitely generated) right R-module is retractable. Some comparisons between max rings, semiartinian rings, perfect rings, noetherian rings, nonsingular rings, and mod-retractable rings are investigated. In particular, we prove ring-theoretical criteria of right mod-retractability for classes of all commutative, left perfect, and right noetherian rings.

When Products of Self-Small Modules are Self-Small

A module M is called “self-small” if the functor Hom(M, −) commutes with direct sums of copies of M. The main goal of the present article is to construct a non-self-small product of self-small modules without nonzero homomorphisms between distinct ones and to correct an error in a claim about products of self-small modules published by Arnold and Murley in a fundamental article on this topic. The second part of the article is devoted to the study of endomorphism rings of self-small modules.

Algebraic properties of word equations

Abstract The question about maximal size of independent system of word equations is one of the most striking problems in combinatorics on words. Recently, Aleksi Saarela has introduced a new approach to the problem that is based on linear-algebraic properties of polynomials encoding the equations and their solutions. In this paper we develop further this approach and take into account other algebraic properties of polynomials, namely their factorization. This, in particular, allows to improve the bound for the number of independent equations with maximal rank from quadratic to linear.

On type-ADS modules and rings

In this paper, we introduce type absolute direct summand (type-ADS) modules and rings as a natural generalization of ADS ones. Besides basic properties and characterizations of the notion, we present several examples illustrating borders of the theory. We also show that some particular classical classes of rings, such as commutative or right non-singular rings are type-ADS.

On socle chains of semiartinian rings with primitive factors artinian

The dimension sequence forms an invariant describing semisimple slices of regular semiartinian rings with primitive factors artinian. Several necessary conditions on dimension sequences are proved under assumption GCH in the paper.

The Defect Functor of a Homomorphism and Direct Unions

We will study commuting properties of the defect functor Defß=CokerHomC(ß,-)

COMMUTATIVE TALL RINGS

\text {Def}_{\beta }=\text {Coker}\text {Hom}_{\mathcal {C}}(\beta ,-)

Small Modules over Abelian Regular Rings

associate to a homomorphism ß in a finitely presented category. As an application, we characterize objects M such that ExtC1(M,-)

Steady ideals and rings

\text {Ext}^{1}_{\mathcal {C}}(M,-)

When every self-small module is finitely generated

commutes with direct unions (i.e. direct limits of monomorphisms), assuming that C