a topoframe, denoted by $l_{ tau}$,  is a pair $(l, tau)$ consisting of a frame $l$ and a subframe $ tau $ all of whose elements are complementary elements in$l$. in this paper, we define and study the notions of a$tau $-real-continuous function on a frame $l$ and the set of realcontinuous functions $mathcal{r}l_tau $ as an $f$-ring.we show that $mathcal{r}l_{ tau}$is actually a generalization ...

Let R be a commutative ring with unity, M unitary R-module and G finite abelian group (viewed as \({\mathbb {Z}}\)-module). The main objective of this paper is to study properties mod-annihilators M. For \(x \in M\), we the ideals \([x : M] =\{r\in ~|~ rM\subseteq Rx\}\) corresponding mod-annihilator We investigate when [x an essential ideal R. prove that arbitrary intersection represented by i...

Introduction Let G be a simply connected and connected semisimple group over an algebraically closed field k of characteristic p > 0. T ⊂ G is a maximal torus and R is the root system relative to T . X(T ) is the weight lattice. Let B ⊃ T be a Borel subgroup corresponding to the negative roots R− = R. Denote by Gr the r-th Frobenius kernel of G. The socle and radical structures of the cohomolog...

Abstract The class of abelian p -groups with minimal full inertia, that is, satisfying the property fully inert subgroups are commensurable invariant is investigated, as well groups not this property; it known both direct sums cyclic and torsion-complete first type. It proved “small" endomorphism ring do satisfy concrete examples them provided via Corner’s realization theorems. Closure properti...

In our paper referred to above we claim to enumerate all …nite primitive linear groups of prime degree r over C with a nonabelian socle. However, the case where the socle is imprimitive was overlooked. In the present paper we deal with this case to complete the classi…cation. 2000 Mathematics Subject Classi…cation: 20H20 20C15 20C33 In the paper referred to above, we state a theorem (Theorem 1....

We prove a version of Hrushovski’s Socle Lemma for rigid groups in an arbitrary simple theory.

Fully invariant subgroups of an Abelian p-group have been the object of a good deal of study, while characteristic subgroups have received somewhat less attention. Recently the socles of fully invariant subgroups have been studied and this led to the notion of a socle-regular group. The present work replaces the fully invariant subgroups with characteristic ones and leads in a natural way to th...

in this paper, we define and study the notion of the real-valued functions on a frame $l$. we show that $f(l) $, consisting of all frame homomorphisms from the power set of $mathbb{r}$ to a frame $ l$, is an $f$-ring, as a generalization of all functions from a set $x$ into $mathbb r$. also, we show that $f(l) $ is isomorphic to a sub-$f$-ring of $mathcal{r}(l)$, the ring of real-valued continu...

Let k be a field and X be a set of n elements. We introduce and study a class of quadratic k-algebras called quantum binomial algebras. Our main result shows that such an algebra A defines a solution of the classical Yang-Baxter equation (YBE), if and only if its Koszul dual A is Frobenius of dimension n, with a regular socle and for each x, y ∈ X an equality of the type xyy = αzzt, where α ∈ k...