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, ..., 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.
• The category of finitely generated projective $R$-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

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