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 establish a topological duality for bounded lattices. The two main features of our duality are that it generalizes Stone duality for bounded distributive lattices, and that the morphisms on either side are not the standard ones. A positive consequence of the choice of morphisms is that those on the topological side are functional. Towards obtaining the topological duality, we develop a unive...

Given a family of Abelian varieties over a positive-dimensional base, we prove that for a sufficiently general curve in the base, every rational section of the family over the curve is contained in a unique rational section over the entire base. 1. Main results sec-intro The starting point for this article is the following theorem. Theorem 1.1. [GHMS05, Theorem 6.2] Let B be a smooth, quasi-pro...

We introduce the class of “right almost V-rings” which is properly between the classes of right V-rings and right good rings. A ring R is called a right almost V-ring if every simple R-module is almost injective. It is proved that R is a right almost V-ring if and only if for every R-module M, any complement of every simple submodule of M is a direct summand. Moreover, R is a right almost V-rin...

In a recent paper [5], we introduced a classes of derived equivalences called almost ν-stable derived equivalences. The most important property is that an almost ν-stable derived equivalence always induces a stable equivalence of Morita type, which generalizes a well-known result of Rickard: derived-equivalent self-injective algebras are stably equivalent of Morita type. In this paper, we shall...