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.

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.

In a previous paper, we investigated the relation between the initial algebra and terminal coalgebra for an endofunctor on the category of sets. In this one we study conditions on a functor to be algebraically compact, which means that the canonical comparison morphism between those objects is an isomorphism. Introduction Suppose C is a category and T : C −→ C is a functor. In both [Barr, 1991]...

Let G be a compact Lie group and M a closed smooth G manifold. With some restrictions on G or on the isotropy groups of the action we show that M is algebraically realized, i. e., there exists a non-singular real algebraic G variety which is equivariantly diffeomorphic to M .

Abstract. This course introduces classical and new results on the algebraic structure of the identity component of the diffeomorphism group Diff0(M) or homeomorphism group Homeo0(M) of a compact manifold. These groups are algebraically simple (no nontrivial normal subgroups) – for deep topological reasons due to Epstein, Mather, Thurston... but nevertheless have a very rich algebraic structure....