It is proved that every commutative ring whose RD-injective modules are Σ-RD-injective is the product of a pure semi-simple ring and a finite ring. A complete characterization of commutative rings for which each artinian (respectively simple) module is RD-injective, is given. These results can be obtained by using the properties of RD-flat modules and RD-coflat modules which are respectively th...

(1) For a function f : A→ B, we call A the domain, B the co-domain, f(A) the range, and f−1(b), for any b ∈ B, the fiber (or inverse image or pre-image) of b. For a subset S of B, f−1(B) = ⋃ b∈S f −1(b). (2) The sizes of fibers can be used to characterize injectivity (each fiber has size at most one), surjectivity (each fiber is non-empty), and bijectivity (each fiber has size exactly one). (3)...

We give a short proof of the theorem that a closed subgroup of a countable product of second countable Lie groups is pro-Lie. The point of this note is to give a short and self-contained, modulo well known results, proof of a theorem of Hofmann and Morris [3] (see also [4, Theorem 3.35] and [5]) in the case of second countable groups. Another simple proof of the result of Hofmann and Morris was...

in this paper‎, ‎we introduce the notion of $(m,n)$-‎algebr‎aically compact modules as an analogue of algebraically‎ ‎compact modules and then we show that $(m,n)$-algebraically‎ ‎compactness‎ ‎and $(m,n)$-pure injectivity for modules coincide‎. ‎moreover‎, ‎further characterizations of a‎ ‎$(m,n)$-pure injective module over a commutative ring are given‎.

an r-module m is called strongly noncosingular if it has no nonzero rad-small (cosingular) homomorphic image in the sense of harada. it is proven that (1) an r-module m is strongly noncosingular if and only if m is coatomic and noncosingular; (2) a right perfect ring r is artinian hereditary serial if and only if the class of injective modules coincides with the class of (strongly) noncosingula...

We find a bound for the Goldie dimension of hereditary modules in terms of the cardinality of the generator sets of its quasi-injective hull. Several consequences are deduced. In particular, it is shown that every right hereditary module with countably generated quasi-injective hull is noetherian. Or that every right hereditary ring with finitely generated injective hull is artinian, thus answe...