Introduction. In this paper we give a sufficient condition that an algebra have a minimal left (or right) ideal. Specifically, we prove that if A is a complex semisimple Banach algebra with the property that the spectrum of every element in A is at most countable, then A has a minimal left ideal. If A is an ^4*-algebra, we prove that A has a minimal left ideal if the spectrum of every self-adjo...

We describe the prime and primitive spectra for quantized enveloping algebras at roots of 1 in characteristic zero in terms of the prime spectrum of the underlying enveloping algebra. Our methods come from the theory of Hopf algebra crossed products. For primitive ideals we obtain an analogue of Duflo’s Theorem, which says that every primitive ideal is the annihilator of a simple highest weight...

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...

If N is a submodule of the R-module M , and a ∈ R, let λa : M/N → M/N be multiplication by a. We say that N is a primary submodule of M if N is proper and for every a, λa is either injective or nilpotent. Injectivity means that for all x ∈ M , we have ax ∈ N ⇒ x ∈ N . Nilpotence means that for some positive integer n, aM ⊆ N , that is, a belongs to the annihilator of M/N , denoted by ann(M/N). ...

Let R be a non-domain commutative ring with identity and A(R) be theset of non-zero ideals with non-zero annihilators. We call an ideal I of R, anannihilating-ideal if there exists a non-zero ideal J of R such that IJ = (0).The annihilating-ideal graph of R is defined as the graph AG(R) with the vertexset A(R) and two distinct vertices I and J are adjacent if and only if IJ =(0). In this paper,...

We introduce a new variant of tight closure associated to any fixed ideal a, which we call a-tight closure, and study various properties thereof. In our theory, the annihilator ideal τ(a) of all a-tight closure relations, which is a generalization of the test ideal in the usual tight closure theory, plays a particularly important role. We prove the correspondence of the ideal τ(a) and the multi...

A ring R is called a left APP-ring if the left annihilator lR(Ra) is right s-unital as an ideal of R for any element a ∈ R. We consider left APP-property of the skew formal power series ring R[[x;α]] where α is a ring automorphism of R. It is shown that if R is a ring satisfying descending chain condition on right annihilators then R[[x;α]] is left APP if and only if for any sequence (b0, b1, ....

Permutation modules are fundamental in the representation theory of symmetric groups Sn and their corresponding Iwahori–Hecke algebras H = H (Sn). We find an explicit combinatorial basis for the annihilator of a permutation module in the “integral” case — showing that it is a cell ideal in G.E. Murphy’s cell structure of H . The same result holds whenever H is semisimple, but may fail in the no...