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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.
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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.
A ring is directly irreducible if and only if 0 and 1 are the only central idempotents. A ring R can be written as e 1 R ⊕ e 2 R ⊕ ... ⊕ e n R with each e i a local idempotent if and only if R is a semiperfect ring. A ring is called an SBI ring or Lift/rad ring if all idempotents of R lift modulo the Jacobson radical.
Let A be a commutative Noetherian ring with unity. Then the following are equivalent. A is Artinian.; A is a finite product of commutative Artinian local rings. [5]A / nil(A) is a semisimple ring, where nil(A) is the nilradical of A.