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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 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.
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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 ...
A secret decoder ring (or secret decoder) is a device that allows one to decode a simple substitution cipher—or to encrypt a message by working in the opposite direction. [ 1 ] As inexpensive toys, secret decoders have often been used as promotional items by retailers, as well as radio and television programs, from the 1930s through to the ...
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Several subtypes are included under this category, the names of which are sometimes used synonymously for the group: wire puzzles; nail puzzles; ring-and-string puzzles; et al. [ 2 ] [ 3 ] Although the initial object is disentanglement, the reverse problem of reassembling the puzzle can be as hard as—or even harder than—disentanglement.
Jacobson himself has defined a ring to be "semisimple" if and only if it is a subdirect product of simple rings, (Jacobson 1989, p. 203). However, this is a stricter notion, since the endomorphism ring of a countably infinite dimensional vector space is semiprimitive, but not a subdirect product of simple rings, ( Lam 1995 , p. 42).