The blob algebra is a finite-dimensional quotient of the Hecke type B which almost always quasi-hereditary. We construct indecomposable tilting modules for over field characteristic 0 in doubly critical case. Every module maximal highest weight either projective or an extension simple by module. Moreover, every submodule weight. conclude that graded Weyl filtration multiplicities this case are ...

Let Δ be an abelian group. By considering the notion multiplication of Δ-graded modules (see [7]) over a commutative Δ-graded ring with unity, we introduce the notion of product of two Δ-graded submodules which we use to characterize the Δ-graded prime submodules of a multiplication Δ-graded module. Finally we proved a graded version of Nakayama lemma for multiplication Δ-graded modules.

(1.2) PV~V where p is defined by p(s, t) = (t, s), (s, tET). We shall assume throughout that T is arcwise connected. Let 77 be an associative and anticommutative graded if-algebra with unit 1, where ii is a field. We assume throughout that 77'= 0 if i<0, and 77° = 7C-1. Let 77+ denote the submodule spanned by the elements of positive degree. 77 is a Hopf algebra over K if there is an algebra ho...

Our effort towards the attainment of high performance devices has yielded several devices with total-area conversion efficiencies above 16%, the highest measuring 16.8% under standard reporting conditions (ASTM E892-87, Global 1000 W/m’). The first attempts to translate this development to larger areas resulted in an efficiency of 12.5% for a 16.8~cm2 monolithically interconnected submodule tes...

the submodules with the property of the title ( a submodule $n$ of an $r$-module $m$ is called strongly dense in $m$, denoted by $nleq_{sd}m$, if for any index set $i$, $prod _{i}nleq_{d}prod _{i}m$) are introduced and fully investigated. it is shown that for each submodule $n$ of $m$ there exists the smallest subset $d&apos;subseteq m$ such that $n+d&apos;$ is a strongly dense submodule of $m$...

Let $M$ be a module over a commutative ring $R$ and let $N$ be a proper submodule of $M$. The total graph of $M$ over $R$ with respect to $N$, denoted by $T(Gamma_{N}(M))$, have been introduced and studied in [2]. In this paper, A generalization of the total graph $T(Gamma_{N}(M))$, denoted by $T(Gamma_{N,I}(M))$ is presented, where $I$ is an ideal of $R$. It is the graph with all elements of $...

1 Definitions Definition 1. A graded ring is a ring S together with a set of subgroups Sd, d ≥ 0 such that S = ⊕ d≥0 Sd as an abelian group, and st ∈ Sd+e for all s ∈ Sd, t ∈ Se. One can prove that 1 ∈ S0 and if S is a domain then any unit of S also belongs to S0. A homogenous ideal of S is an ideal a with the property that for any f ∈ a we also have fd ∈ a for all d ≥ 0. A morphism of graded r...

Let R be a Noetherian ring, F := Rr and M ⊆ F a submodule of rank r. Let A∗(M) denote the stable value of Ass(Fn/Mn), for n large, where Fn is the nth symmetric power of Fn and Mn is the image of the nth symmetric power of M in Fn. We provide a number of characterizations for a prime ideal to belong to A∗(M). We also show that A∗(M) ⊆ A∗(M), where A∗(M) denotes the stable value of Ass(Fn/Mn).

Let G be a group and R G-graded ring. In this paper, we present examine the concept of graded weakly 2-absorbing ideals as in generality prime ring which is not commutative, demonstrates that symmetry obtained lot outcomes commutative rings remain are commutative.