On injectivity and p-injectivity. (English) Zbl 0655.16012

Yue Chi Ming, Roger

J. Math. Kyoto Univ. 27, 439-452 (1987).

From the author’s introduction: “In this paper, certain results known for semiprime left Goldie rings are shown to have their analogs for rings with von Neumann regular classical left quotient rings. Conditions for von Neumann regularity and connections between injective modules over certain rings are studied. The following are among the results proved here: (1) Let A be a ring having a classical left quotient ring Q. Then (a) if A is left hereditary and Q regular, then A is left Noetherian iff every essential left ideal of A is essentially finitely generated; (b) Q is regular iff every divisible torsionfree left A-module is p-injective; (c) Q is semisimple Artinian iff every divisible torsionfree quasi- injective left A-module is injective; (2) The following conditions are equivalent: 1) A is von Neumann regular; 2) A has a classical left quotient ring which is a projective left A-module and every finitely generated divisible singular left A-module is flat; 3) for any non-zero proper principal right ideal I of A, there exist a non-trivial idempotent e and a left regular element d such that \(I=edA\); (3) Let A be a left p.p. ring (every principal left ideal is projective) having a classical left quotient ring Q. If Q is left p-injective then it is von Neumann regular; (4) Let A be a left YJ-ring with Jacobson radical J and satisfying the maximum condition on left annihilators. If E, Q are injective left A-modules such that \(r_ Q(J)\) is isomorphic to \(r_ E(J)\) (as left A-modules), then Q is isomorphic to E.”

Reviewer: S.K.Jain

MSC:

16D50	Injective modules, self-injective associative rings
16E50	von Neumann regular rings and generalizations (associative algebraic aspects)
16P50	Localization and associative Noetherian rings

Keywords:

von Neumann regular classical left quotient rings; injective modules; divisible torsionfree quasi-injective left A-module; finitely generated divisible singular left A-module; left p.p. ring; left annihilators