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.

Contents 1 Definition 2 Examples 3 Properties 4 References

[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, ..., e_n of idempotents with each e_i R e_i 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.