In this work, we introduce $H^*$-condition on the set of submodules of a module. Let $M$ be a module. We say $M$ satisfies $H^*$ provided that for every submodule $N$ of $M$, there is a direct summand$D$ of $M$ such that $(N+D)/N$ and $(N+D)/D$ are cosingular. We show that over a right perfect right $GV$-ring,a homomorphic image of a $H^*$ duo module satisfies $H^*$.

An irreducible representation of a simple Lie algebra can be a direct summand of its own tensor square. In this case, the representation admits a nonassociative algebra structure which is invariant in the sense that the Lie algebra acts as derivations. We study this situation for the Lie algebra sl(2).

We compare the domain of the assembly map in algebraic K – theory with respect to the family of finite subgroups with the domain of the assembly map with respect to the family of virtually cyclic subgroups and prove that the former is a direct summand of the later. AMS Classification 19D50; 19A31, 19B28

A Warreld group is a direct summand of a simply presented abelian group. In this paper, we describe the Pon-trjagin dual groups of Warreld groups, both for the p-local and the general case. A variety of characterizations of these dual groups is obtained. In addition, numerical invariants are given that distinguish between two such groups which are not topologically isomorphic.

a module is said to be $pi$-extending provided that every projection invariant submodule is essential in a direct summand of the module. in this paper, we focus on direct summands and indecomposable decompositions of $pi$-extending modules. to this end, we provide several counter examples including the tangent bundles of complex spheres of dimensions bigger than or equal to 5 and certain hyper ...

We discuss sharpness in the Hausdorff Young theorem for unimodular groups. First the functions on unimodular locally compact groups for which equality holds in the Hausdorff Young theorem are determined. Then it is shown that the Hausdorff Young theorem is not sharp on any unimodular group which contains the real Une as a direct summand, or any unimodular group which contains an Abelian normal ...

We show that the closure of compactly supported mapping class group an infinite type surface is not perfect and its abelianization contains a direct summand isomorphic to uncountable sum rationals. also extend this Torelli in case surfaces with genus indivisible copy free abelian as well. Finally we give application question automatic continuity by exhibiting discontinuous homomorphisms