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Dually slender modules and steady rings. (English) Zbl 0866.16003
Let $$R$$ be an associative ring with identity. A (right) $$R$$-module $$M$$ is called dually slender (or small) whenever $$\text{Hom}_R(M,-)$$ commutes with direct sums. Every finitely generated module is dually slender but the converse does not hold in general. The rings such that every dually slender right $$R$$-module is finitely generated are called right steady and it was shown by J. Trlifaj [Abelian groups and modules, Padova 1994, 467-473 (1995; Zbl 0845.16009)] that every right semiartinian ring of finite Loewy length is right steady. In the paper under review this result is generalized to right semiartinian rings of countable Loewy length, which also turn out to be right steady. The authors show that in the case of uncountable Loewy length the situation is much more complicated. They construct semiartinian steady rings of arbitrary non-limit Loewy length and also semiartinian non-steady rings of arbitrary uncountable non-limit Loewy length.
The last sections of the paper are devoted to the study of steady rings by means of model-theoretic methods. The authors obtain a model-theoretic necessary and sufficient condition for a countable ring to be steady. The paper ends with a study of the relations among dually slender, reducing, and almost free modules.

##### MSC:
 16D70 Structure and classification for modules, bimodules and ideals (except as in 16Gxx), direct sum decomposition and cancellation in associative algebras) 16P20 Artinian rings and modules (associative rings and algebras) 16D40 Free, projective, and flat modules and ideals in associative algebras 16D90 Module categories in associative algebras 16E50 von Neumann regular rings and generalizations (associative algebraic aspects) 03C60 Model-theoretic algebra 16B70 Applications of logic in associative algebras 16S15 Finite generation, finite presentability, normal forms (diamond lemma, term-rewriting)
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