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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.
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 ...
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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.
The inclusion functor Ring → Rng has a left adjoint which formally adjoins an identity to any rng. The inclusion functor Ring → Rng respects limits but not colimits. The zero ring serves as both an initial and terminal object in Rng (that is, it is a zero object). It follows that Rng, like Grp but unlike Ring, has zero morphisms. These are ...