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 show that finite-dimensional Lie algebras over a field of characteristic zero such that the second cohomology group in every finite-dimensional module vanishes, are, essentially, semisimple.

We introduce the notions of T-dual Rickart and strongly T-dual Rickart modules. We provide several characterizations and investigate properties of each of these concepts. It is shown that every free (resp. finitely generated free) $R$-module is T-dual Rickart if and only if $overline{Z}^2(R)$ is a  direct summand of $R$ and End$(overline{Z}^2(R))$ is a semisimple (resp. regular) ring. It is sho...

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...

For a linear semisimple Lie group we obtain a necessary and sufficient condition for a highest weight module with non-singular infinitesimal character to be unitarizable.

We show that finite-dimensional Lie algebras over a field of characteristic zero such that the second cohomology group in every finite-dimensional module vanishes, are, essentially, semisimple.

We recall a version of the Osofsky-Smith theorem in the context of a Grothendieck category and derive several consequences of this result. For example, it is deduced that every locally finitely generated Grothendieck category with a family of completely injective finitely generated generators is semisimple. We also discuss the torsion-theoretic version of the classical Osofsky theorem which cha...

Permutation modules are fundamental in the representation theory of symmetric groups Sn and their corresponding Iwahori–Hecke algebras H = H (Sn). We find an explicit combinatorial basis for the annihilator of a permutation module in the “integral” case — showing that it is a cell ideal in G.E. Murphy’s cell structure of H . The same result holds whenever H is semisimple, but may fail in the no...

Let H be an involutory Hopf algebra over a field of characteristic zero, M and N two finite dimensional left H-modules such that M ⊗ N is a semisimple H-module. Then M and N are semisimple H-modules. This is a generalization of a theorem proved by J.-P. Serre for group algebras. A version of the theorem above for monoidal categories is also given.