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The following equivalent definitions of a left perfect ring R are found in Anderson and Fuller: [2]. Every left R-module has a projective cover.; R/J(R) is semisimple and J(R) is left T-nilpotent (that is, for every infinite sequence of elements of J(R) there is an n such that the product of first n terms are zero), where J(R) is the Jacobson radical of R.
Let C be an additive category, or more generally an additive R-linear category for a commutative ring R. We call C a Krull–Schmidt category provided that every object decomposes into a finite direct sum of objects having local endomorphism rings. Equivalently, C has split idempotents and the endomorphism ring of every object is semiperfect.
A ring R is left self-injective if the module R R is an injective module. While rings with unity are always projective as modules, they are not always injective as modules. semiperfect A semiperfect ring is a ring R such that, for the Jacobson radical J(R) of R, (1) R/J(R) is semisimple and (2) idempotents lift modulo J(R). semiprimary
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The classical ring of quotients for any commutative Noetherian ring is a semilocal ring. The endomorphism ring of an Artinian module is a semilocal ring. Semi-local rings occur for example in algebraic geometry when a (commutative) ring R is localized with respect to the multiplicatively closed subset S = ∩ (R \ p i ) , where the p i are ...
Semisimple rings are left and right hereditary via the equivalent definitions: all left and right ideals are summands of R, and hence are projective.By a similar token, in a von Neumann regular ring every finitely generated left and right ideal is a direct summand of R, and so von Neumann regular rings are left and right semihereditary.