We investigate preserving of projectivity and injectivity by the object-wise tensor product of RC-modules, where C is a small category. In particular, let O(G,X) be the category of canonical orbits of a discrete group G, over a G-set X. We show that projectivity of RO(G,X)-modules is preserved by this tensor product. Moreover, if G is a finite group, X a finite G-set and R is an integral domain...

In this paper certain injectivity conditions in terms of extensions of monomorphisms are considered. In particular, it is proved that a ring R is a quasi-Frobenius ring if and only if every monomorphism from any essential right ideal of RR into R (N) R can be extended to RR. Also, known results on pseudo-injective modules are extended. Dinh raised the question if a pseudo-injective CS module is...

Proof: For f ∈ Hom(X,A), i ◦ f = 0 implies (i ◦ f)(x) = 0 for all x ∈ X, and then f(x) = 0 for all x since i is injective. Thus, Hom(X,A)→ Hom(X,B) is injective, giving exactness at the left joint. Since q ◦ i = 0, any f ∈ Hom(X,A) is mapped to 0 ∈ Hom(X,C) by f → q ◦ i ◦ f . That is, the image of i ◦ − is contained in the kernel of q ◦ −. On the other hand, when g ∈ Hom(X,B) is mapped to q ◦ g...