Preinjective modules and finite representation type of artinian rings
作者:
Nguyen Viet dung,
期刊:
Communications in Algebra
(Taylor Available online 1999)
卷期:
Volume 27,
issue 8
页码: 3921-3947
ISSN:0092-7872
年代: 1999
DOI:10.1080/00927879908826674
出版商: Gordon and Breach Science Publishers Ltd.
数据来源: Taylor
摘要:
LetRbe a left artinian ring such that every finitely presented right .ft-module is of finite endolength. It is shown that the cardinality of the set of isomorphism classes of preinjective rightR-modules is less than or equal to the cardinality of the set of isomorphism classes of preprojective leftR-modules, andRis of finite representation type if and only if these cardinal numbers are finite and equal to each other. As a consequence, we deduce a theorem, due to Herzog [17], asserting that a left pure semisimple ringRis of finite representation type if and only if the number of non-isomorphic preinjective rightR-modules is the same as the number of non-isomorphic preprojective left .R-modules. Further applications are also given to provide new criteria for artinian rings with self-duality and artinian Pi-rings to be of finite representation type, which imply in particular the validity of the pure semisimple conjecture for these classes of rings.
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