CHARACTERIZATIONS OF NOETHERIAN AND ARTINIAN RINGS Mai Duy Tan
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Abstract
For the last few decades, characterizing rings in terms of finitely generated modules has become a topic of research for many algebraists. Let R be a ring, the aim of this paper is to prove the theorems: R is right noetherian (respectively artinian) if and only if every 2-generated right R-module is noetherian (respectively artinian) or an ADS module. Using this result, we proved that a ring R is semi-simple if and only if every 2-generated R-module is ADS. In addition, we also proved that a ring R is right SC if and only if every 2-generated R−module is ADS. This result generalizes a theorem of Rizvi which states that a ring R is right SC if and only if every finitely generated right R-module is quasi-continuous.
Keywords
artinian, modules, noetherian, rings
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References
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