Vol. 59, No. 1, 1975

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On M-projective and M-injective modules

Goro Azumaya, F. Mbuntum and Kalathoor Varadarajan

Vol. 59 (1975), No. 1, 9–16

In this paper necessary and sufficient conditions are obtained for a direct sum αJAα of R-modules to be M. injective in the sense of Azumaya. Using this result it is shown that if {Aα}αJ is a family of R-modules with the property that αKAα is M-injective for every countable subset K of J then αJAα is itself M-injective. Also we prove that arbitrary direct sums of M-injective modules are M-injective if and only if 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 if 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.

Mathematical Subject Classification
Primary: 16A52
Received: 19 June 1974
Published: 1 July 1975
Goro Azumaya
F. Mbuntum
Kalathoor Varadarajan