Some Comments on Injectivity and P-injectivity

A generalization of injective modules (noted GI-modules), distinct from p-injective modules, is introduced. Rings whose p-injective modules are GI are characterized. If M is a left GI-module, E = End(AM), then E/J(E) is von Neumann regular, where J(E) is the Jacobson radical of the ring E. A is semisimple Artinian if, and only if, every left A-module is GI. If A is a left p. p., left GI-ring such that every non-zero complement left ideal of A contains a non-zero ideal of A, then A is strongly regular. Sufficient conditions are given for a ring to be either von Neumann regular or quasi-Frobenius. Quasi-Frobenius and von Neumann regular rings are characterized. Kasch rings are also considered. Throughout, A denotes an associative ring with identity and A-modules are unital. J , Z, Y will stand respectively for the Jacobson radical, the left singular ideal and the right singular ideal of A. A is called semi-primitive or semi-simple (resp. (a) left non-singular; (b) right non-singular) if J = (0) (resp. (a) Z = (0); (b) Y = (0)). An ideal of A will always mean a two-sided ideal of A. A is called left (resp. right) quasi-duo if every maximal left (resp. right) ideal of A is an ideal of A. It well-known that J , Z, Y are ideals of A. A left (right) ideal of A is called reduced if it contains no non-zero nilpotent elements. Following C. Faith, write “A is VNR” if A is a von Neumann regular ring [8]. A is called fully (resp. (1) fully left; (2) fully right) idempotent if every ideal (resp.(1) left ideal ; (2) right ideal) of A is idempotent. It is well-known that A is VNR if and only if every left (right) A-module is flat (Harada ((1956); Auslander (1957)). Also, A is VNR if and only if every left (right) A-module is p-injective ([2], [4], [12], [22], [23]). Note that the Harada-Auslander’s characterization may be weakened as follows: A is VNR if and only if every singular right A-module is flat (cf. [38, p. 147]). Recall that a left A-module M is p-injective if, for any principal left ideal P of A, every left A-homomorphism of P into M extends to one of A into M ([8, p. 122], [20, p. 577], [21, p. 340], [26]). A is called a left p-injective ring if AA is p-injective. P-injectivity is similarly defined on the right side. A generalization of p-injectivity, noted YJ-injectivity, is introduced in [29](cf. also [22], [39]). YJ-injectivity is also called GP-injectivity by other authors (cf. [4], [6], [15]). Received December 16, 2005. 2000 Mathematics Subject Classification. Primary 16D40, 16D50, 16E50.