Semiperfect ring
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In abstract algebra, a semiperfect ring is a ring over which every finitely generated left module has a projective cover. This property is left right symmetric.
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[edit] Definition
Let R be ring. Then R is semiperfect if any of the following equivalent conditions hold:
- R/J(R) is semisimple and idempotents lift modulo J(R), where J(R) is the Jacobson radical of R.
- R has a complete orthogonal set e1, ..., en of idempotents with each ei R ei a local ring.
- Every simple left (right) R-module has a projective cover.
- Every finitely generated left (right) R-module has a projective cover.
[edit] Examples
Examples of semiperfect rings include:
- Perfect rings.
- Local rings.
- Left (right) Artinian rings.
- Finite dimensional k-algebras.
[edit] Properties
Since a ring R is semiperfect iff every simple left R-module has a projective cover, every ring Morita equivalent to a semiperfect ring is also semiperfect.
[edit] References
- Anderson, Frank Wylie; Fuller, Kent R (1992). Rings and Categories of Modules. Springer. ISBN 0387978453. Retrieved on 2007-03-27.
