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.
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.
- The category of finitely generated projective
-modules is Krull-Schmidt.
Examples
Examples of semiperfect rings include:
- Left (right) perfect rings.
- Local rings.
- Left (right) Artinian rings.
- Finite dimensional k-algebras.
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.
References
- Anderson, Frank Wylie; Fuller, Kent R (1992). Rings and Categories of Modules. Springer. ISBN 0-387-97845-3. Retrieved 2007-03-27.
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