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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.
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.
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
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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A semiperfect number is necessarily either perfect or abundant. An abundant number that is not semiperfect is called a weird number. With the exception of 2, all primary pseudoperfect numbers are semiperfect. Every practical number that is not a power of two is semiperfect. The natural density of the set of semiperfect numbers exists. [2]
A semiperfect number is a natural number that is equal to the sum of all or some of its proper divisors. A semiperfect number that is equal to the sum of all its proper divisors is a perfect number. Most abundant numbers are also semiperfect; abundant numbers which are not semiperfect are called weird numbers.