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Dually slender modules and steady rings

Dually slender modules and steady rings Abstract. A module M over a ring R is called dually slender if Hom (M, --) commutes with direct sums of -modules. For example, any finitely generated module is dually slender. A ring R is called right steady if each dually slender right -module is finitely generated. We provide a model theoretic necessary and sufficient condition for a countable ring to be right steady. Also, we prove that any right semiartinian ring of countable Loewy length is right steady. For each uncountable ordinal , we construct examples of commutative semiartinian rings , and Qa, of Loewy length +1 such that is, but Q0 is not, steady. Finally, we study relations among dually slender, reducing, and almost free modules. 1991 Mathematics Subject Classification: 16D40, 16D90, 16E50, 03C60. Introduction In this note, we study two recent generalizations of the notion of a finitely generated module, namely the notion of a dually slender module, and of a (, o>)-reducing module. Both generalizations share the property that they do not include any countably infmitely generated modules. Dually slender modules are defined exactly s duals of slender modules (see Definition 1.1 and Lemma l .2 below). They have been studied - under various http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Forum Mathematicum de Gruyter

Dually slender modules and steady rings

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References (12)

Publisher
de Gruyter
Copyright
Copyright © 2009 Walter de Gruyter
ISSN
0933-7741
eISSN
1435-5337
DOI
10.1515/form.1997.9.61
Publisher site
See Article on Publisher Site

Abstract

Abstract. A module M over a ring R is called dually slender if Hom (M, --) commutes with direct sums of -modules. For example, any finitely generated module is dually slender. A ring R is called right steady if each dually slender right -module is finitely generated. We provide a model theoretic necessary and sufficient condition for a countable ring to be right steady. Also, we prove that any right semiartinian ring of countable Loewy length is right steady. For each uncountable ordinal , we construct examples of commutative semiartinian rings , and Qa, of Loewy length +1 such that is, but Q0 is not, steady. Finally, we study relations among dually slender, reducing, and almost free modules. 1991 Mathematics Subject Classification: 16D40, 16D90, 16E50, 03C60. Introduction In this note, we study two recent generalizations of the notion of a finitely generated module, namely the notion of a dually slender module, and of a (, o>)-reducing module. Both generalizations share the property that they do not include any countably infmitely generated modules. Dually slender modules are defined exactly s duals of slender modules (see Definition 1.1 and Lemma l .2 below). They have been studied - under various

Journal

Forum Mathematicum – de Gruyter

Published: Jan 1, 1997

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