In this article, we study the restriction of Zuckerman’s derived functor (g,K)-modules Aq(λ) to g′ for symmetric pairs of reductive Lie algebras (g, g′). When the restriction decomposes into irreducible (g′,K′)-modules, we give an upper bound for the branching law. In particular, we prove that each (g′,K′)-module occurring in the restriction is isomorphic to a submodule of Aq′ (λ ′) for a parab...

Let k be an algebraically closed field of characteristic p, possibly zero, and G = qGL3(k), the quantum group of three by three matrices as defined by Dipper and Donkin. We may also take G to be GL3(k). We first determine the extensions between simple G-modules for both G and G1, the first Frobneius kernel of G. We then determine the submodule structure of certain induced modules, Ẑ(λ), for the...

Let W be a finite-dimensional Z/p-module over a field, k, of characteristic p. The maximum degree of an indecomposable element of the algebra of invariants, k[W ]Z/p, is called the Noether number of the representation, and is denoted by β(W ). A lower bound for β(W ) is derived, and it is shown that if U is a Z/p submodule of W , then β(U) 6 β(W ). A set of generators, in fact a SAGBI basis, is...

The modular multilevel converter (MMC) is receiving extensive research interests in high/medium voltage applications due to its modularity, scalability, reliability, high voltage capability and excellent harmonic performance. The submodule capacitors are usually quite bulky since they have to withstand fundamental frequency voltage fluctuations. To reduce the capacitance of these capacitors, th...

let $r$ be an arbitrary ring and $t$ be a submodule of an $r$-module $m$. a submodule $n$ of $m$ is called $t$-small in $m$ provided for each submodule $x$ of $m$, $tsubseteq x+n$ implies that $tsubseteq x$. we study this mentioned notion which is a generalization of the small submodules and we obtain some related results.

The object of this paper is to study the relationship between certain projective modules and their endomorphism rings. Specifically, the basic problem is to describe the projective modules whose endomorphism rings are (von Neumann) regular, local semiperfect, or left perfect. Call a projective module regular if every cyclic submodule is a direct summand. Thus a ring is a regular module if it is...

Let k be a field of positive characteristic and K = k(V ) a function field of a variety V over k and let AK be the ring of adéles of K with respect to the places on K corresponding to the divisors on V . Given a Drinfeld module Φ : F[t] → EndK(Ga) over K and a positive integer g we regard both K and AgK as Φ(Fp[t])-modules under the diagonal action induced by Φ. For Γ ⊆ K a finitely generated Φ...

The purpose of this paper is to investigate pure submodules of multiplication modules. We introduce the concept of idempotent submodule generalizing idempotent ideal. We show that a submodule of a multiplication module with pure annihilator is pure if and only if it is multiplication and idempotent. Various properties and characterizations of pure submodules of multiplication modules are consid...

The purpose of this paper is to introduce a new concept in module M over ring R, called e∗-essential submodule, which generalization an essential submodule. We will some examples and properties about such that, what the inverse image intersection submodules direct sum submodules. show relationship between submodule Noetherian R-module. Also we define e∗-closed with

Let K be a field, K̄ its separable closure, Gal(K) = Gal(K̄/K) the (absolute) Galois group of K. Let X be an abelian variety over K. If n is a positive integer that is not divisible by char(K) then we write Xn for the kernel of multiplication by n in X(Ks). It is well-known [21] that Xn ia a free Z/nZ-module of rank 2dim(X); it is also a Galois submodule in X(K̄). We write K(Xn) for the field of d...