On M-projective and M-injective modules

Abstract

In this paper necessary and sufficient conditions are obtained for a direct sum ⊕α∈J Aα of R-modules to be M-injective in the sense of Azumaya. Using this result it is shown that H {Aα}α∈J is a family of R-modules with the property that A is M-injective for every countable subset K of J then ⊕∝εJ is itself M-injective. Also we prove that arbitrary direct sums of M-injective modules are M-injective H and only M is locally noetherian, in the sense that every cyclic submodule of M is noetherian. We also obtain some structure theorems about Z-projective modules in the sense of Azumaya, where Z denotes the ring of integers. Writing any abelian group A as D ⊕ H with D divisible and H reduced, we show that H A is Z-projective then H is torsion free and every pure subgroup of finite rank of H is a free direct summand of H. © 1975 Pacific Journal of Mathematics.