Ads
related to: semiperfect ringjamesallen.com has been visited by 10K+ users in the past month
Search results
Results From The WOW.Com Content Network
The right case is defined by analogy, and the condition is not left-right symmetric; that is, there exist rings which are perfect on one side but not the other. Perfect rings were introduced in Bass's book. [1] A semiperfect ring is a ring over which every finitely generated left module has a projective cover. This property is left-right symmetric.
The projective indecomposable modules over some rings have very close connections with those rings' simple, projective, and indecomposable modules. If the ring R is Artinian or even semiperfect , then R is a direct sum of principal indecomposable modules, and there is one isomorphism class of PIM per isomorphism class of simple module.
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
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 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.
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 () where P is some finitely generated progenerator B. This is why the results are phrased in terms of ...