Miriam Cohen raised the question whether the smash product of a semisimple Hopf algebra and a semiprime module algebra is semiprime. In this paper we show that the smash product of a commutative semiprime module algebra over a semisimple cosemisimple Hopf algebra is semiprime. In particular we show that the central H-invariant elements of the Martindale ring of quotients of a module algebra for...

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...

Let R be a left pure semisimple ring such that there are no nonzero homomorphisms from preinjective modules to non-preinjective indecomposable modules in R-mod, and let W be the left key R-module; i.e., W is the direct sum of all non-isomorphic non-preinjective indecomposable direct summands of products of preinjective left R-modules. We show that if the module W is endofinite, then R is a ring...

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...

Let R be a commutative ring. An R-module M is called co-multiplication provided that foreach submodule N of M there exists an ideal I of R such that N = (0 : I). In this paper weshow that co-multiplication modules are a generalization of strongly duo modules. Uniserialmodules of finite length and hence valuation Artinian rings are some distinguished classes ofco-multiplication rings. In additio...

In this paper, we define the higher Frobenius-Schur (FS-)indicators for finite-dimensional modules of a semisimple quasi-Hopf algebra H via the categorical counterpart developed in a 2005 preprint. When H is an ordinary Hopf algebra, we show that our definition coincides with that introduced by Kashina, Sommerhäuser, and Zhu. We find a sequence of gauge invariant central elements of H such that...

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...

LetG be a quasisimple, connected, and simply connected algebraic group defined and split over the field k of characteristic p > 0. In this paper, we are interested in small modules for G; for us, small modules are those with dimension ≤ p. By results of Jantzen [Jan96] one knows that anyGmodule V with dimV ≤ p is semisimple. (We always understand aG-module V to be given by a morphism of algebra...

Let H be a semisimple Hopf algebra over an algebraically closed field k. It is assumed that either char k = 0 or char k > dimH . Semisimplicity of H means that H is a semisimple left H-module. In that case H has finite dimension. Throughout the paper we shall keep to notations from [M]. For example H stands for the dual Hopf algebra with the natural pairing 〈−,−〉 : H ⊗ H → k. The algebra H is a...

We propose a new approach for the study of the Jacquet module of a Harish-Chandra module of a real semisimple Lie group. Using this method, we investigate the structure of the Jacquet module of principal series representation generated by the trivial K-type.