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2. Perfect ring - Wikipedia

   en.wikipedia.org/wiki/Perfect_ring

   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.

3. Idempotent (ring theory) - Wikipedia

   en.wikipedia.org/wiki/Idempotent_(ring_theory)

   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.

4. Principal indecomposable module - Wikipedia

   en.wikipedia.org/wiki/Principal_indecomposable...

   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.

5. Kaplansky's theorem on projective modules - Wikipedia

   en.wikipedia.org/wiki/Kaplansky's_theorem_on...

   The theorem can also be formulated so to characterize a local ring (#Characterization of a local ring). For a finite projective module over a commutative local ring, the theorem is an easy consequence of Nakayama's lemma. [3] For the general case, the proof (both the original as well as later one) consists of the following two steps:

6. Krull–Schmidt category - Wikipedia

   en.wikipedia.org/wiki/Krull–Schmidt_category

   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.

7. Could $5,000 DOGE stimulus checks really happen? Here's what ...

   www.aol.com/could-5-000-doge-stimulus-194354650.html

   President Trump has floated the idea of giving taxpayers a $5,000 refund check as a dividend from savings created by DOGE. Here's what to know.