S. Montgomery and S. Witherspoon proved that upper and lower semisolvable, semisimple, finite dimensional Hopf algebras are of Frobenius type when their dimensions are not divisible by the characteristic of the base field. In this note we show that a finite dimensional, semisimple, lower solvable Hopf algebra is always of Frobenius type, in arbitrary characteristic.

in this paper, we show that every surjective $n$-homomorphism ($n$-anti-homomorphism) from a banach algebra $a$ into a semisimple banach algebra $b$ is continuous.

The notion of simple-direct-injective modules which are a generalization injective unifies $C2$ and $C3$-modules. In the present paper, we introduce semisimple-direct-injective module gives unified viewpoint $C2$, $C3$, SSP properties modules. It is proved that ring $R$ Artinian serial with Jacobson radical square zero if only every right $R$-module has and, for any family simple $R$-modules $\...

Let G be an algebraic group over a field k. We call g ∈ G(k) real if g is conjugate to g−1 in G(k). In this paper we study reality for groups of type G2 over fields of characteristic different from 2. Let G be such a group over k. We discuss reality for both semisimple and unipotent elements. We show that a semisimple element in G(k) is real if and only if it is a product of two involutions in ...

In this paper, we consider the characterization of h -semisimple hemirings. First, the notions of ( , ) q ∈∈∨ -fuzzy h -ideals and ( , ) q ∈∈∨ -fuzzy h -interior ideals of a hemiring are introduced, and some of their properties are investigated. Then the concept of h -semisimple hemirings is introduced and its characterizations are established.

We start by introducing Kac-Moody algebras and completing the classification of finite dimensional semisimple Lie algebras. We then discuss the classification of finite dimensional representations of semisimple Lie algebras (and, more generally, integrable highest weight representations of Kac-Moody algebras). We finish by discussing the structure and representation theory of reductive algebrai...