In this paper necessary and
sufficient conditions are obtained for a direct sum ⊕α∈JAα of R-modules to be M.
injective in the sense of Azumaya. Using this result it is shown that if {Aα}α∈J is a
family of R-modules with the property that ⊕α∈KAα is M-injective for every
countable subset K of J then ⊕α∈JAα 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.
