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.

We prove injectivity and relative asphericity for "layered" systems of equations over torsion-free groups, when the exponent matrix is invertible over Z.

In my talk I will discuss the following results which were obtained in joint work with Wilderich Tuschmann. 1. For any given numbers m, C and D, the class of m-dimensional simply connected closed smooth manifolds with finite second homotopy groups which admit a Riemannian metric with sectional curvature |K| ≤ C and diameter ≤ D contains only finitely many diffeomorphism types. 2. Given any m an...

Given a Galois extension $$R^{\beta } \subset R$$ , where $$\beta $$ is an action of finite groupoid on noncommutative ring, we present some conditions to the map be injective.

Given a smooth compact Riemannian surface, we prove that if a suitable convexity assumption on the tangent focal cut loci is satisfied, then all injectivity domains are semiconvex.