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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.
Perfect ring#Semiperfect ring; This page is a redirect. The following categories are used to track and monitor this redirect: To a subtopic: ...
Examples of multiplicative sets include: the set-theoretic complement of a prime ideal in a commutative ring; the set {1, x, x 2, x 3, ...}, where x is an element of a ring; the set of units of a ring; the set of non-zero-divisors in a ring; 1 + I for an ideal I; the Jordan–Pólya numbers, the multiplicative closure of the factorials.
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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