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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.
Also, for semiperfect rings such as serial rings, the basic ring is Morita equivalent to the original ring. Thus if R is a serial ring with basic ring B , and the structure of B is known, the theory of Morita equivalence gives that R ≅ E n d B ( P ) {\displaystyle R\cong \mathrm {End} _{B}(P)} where P is some finitely generated progenerator B .
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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 ...
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The rings () and () are not simple and not Artinian if the set I is infinite, but they are still full linear rings. The Artin–Wedderburn theorem states that every semisimple ring is isomorphic to a finite direct product ∏ i = 1 r M n i ( D i ) {\textstyle \prod _{i=1}^{r}\operatorname {M} _{n_{i}}(D_{i})} , for some nonnegative integer ...
The ring R is called a semiprime ring if the zero ideal is a semiprime ideal. In the commutative case, this is equivalent to R being a reduced ring, since R has no nonzero nilpotent elements. In the noncommutative case, the ring merely has no nonzero nilpotent right ideals. So while a reduced ring is always semiprime, the converse is not true. [1]