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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.
Gilbert cloud chamber, assembled An alternative view of kit contents. The lab contained a cloud chamber allowing the viewer to watch alpha particles traveling at 12,000 miles per second (19,000,000 m/s), a spinthariscope showing the results of radioactive disintegration on a fluorescent screen, and an electroscope measuring the radioactivity of different substances in the set.
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.
DIY was originally founded by Vimeo co-founder Zach Klein, [2] Isaiah Saxon, Andrew Sliwinski, and Daren Rabinovitch in May 2012. [3] [4] The company launched a second online children's educational platform in 2016 called JAM.com, [5] [6] which was subscription-based and more focused on a course structure for learning versus DIY's free and badge-based skill building structure.
A theorem of Hyman Bass in now known as "Bass' Theorem P" showed that the descending chain condition on principal left ideals of a ring R is equivalent to R being a right perfect ring. D. D. Jonah showed in ( Jonah 1970 ) that there is a side-switching connection between the ACCP and perfect rings.
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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 ...