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| {{Merge to |perfect ring |date=March 2011}}
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| In [[abstract algebra]], a '''semiperfect ring''' is a [[ring (mathematics)|ring]] over which every [[finitely generated module|finitely generated]] left [[module (mathematics)|module]] has a [[projective cover]]. This property is left-right symmetric.
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| == Definition ==
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| Let ''R'' be ring. Then ''R'' is '''semiperfect''' if any of the following equivalent conditions hold:
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| * ''R''/J(''R'') is [[semisimple module|semisimple]] and [[idempotent element|idempotent]]s lift modulo J(''R''), where J(''R'') is the [[Jacobson radical]] of ''R''.
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| * ''R'' has a complete orthogonal set ''e''<sub>1</sub>, ..., ''e''<sub>''n''</sub> of idempotents with each ''e''<sub>''i''</sub> ''R e''<sub>''i''</sub> a [[local ring]].
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| * Every [[simple module|simple]] left (right) [[module (mathematics)|''R''-module]] has a [[projective cover]].
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| * Every [[finitely generated module|finitely generated]] left (right) ''R''-module has a projective cover.
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| * The category of finitely generated projective <math>R</math>-modules is [[Krull-Schmidt_category|Krull-Schmidt]].
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| == Examples ==
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| Examples of '''semiperfect rings''' include:
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| * Left (right) [[perfect ring]]s.
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| * [[Local ring]]s.
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| * Left (right) [[Artinian ring]]s.
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| * [[dimension (vector space)|Finite dimensional]] [[algebra over a field|''k''-algebras]].
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| == Properties ==
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| Since a ring ''R'' is semiperfect iff every [[simple module|simple]] left [[module (mathematics)|''R''-module]] has a projective cover, every ring [[Morita equivalence|Morita equivalent]] to a semiperfect ring is also semiperfect.
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| == References ==
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| *{{cite book|last = Anderson|first = Frank Wylie|coauthors = Fuller, Kent R|title = Rings and Categories of Modules|publisher = Springer|date = 1992|isbn = 0-387-97845-3|url = http://books.google.com/books?id=PswhrD_wUIkC|accessdate = 2007-03-27}}
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| [[Category:Ring theory]]
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| [[Category:Module theory]]
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