Goldie’s Theorem (1960), which is one of the most important results in Ring Theory, is a criterion for a ring to have a semisimple left quotient ring. The aim of the paper is to give four new criteria (using a completely different approach and new ideas). The first one is based on the recent fact that for an arbitrary ring R the set M of maximal left denominator sets of R is a non-empty set [2]...

0 Introduction This is a reasonably faithful account of the ve lectures I delivered at the summer course \Geometria Algebraica no Commutativa y Espacios Cuanti-cos" for graduate students, in Spain, July 25{29, 1994. The material covered was, for the most part, an abridged version of Artin and Zhang's paper 2]. Fix a eld k. Given a Z-graded k-algebra, A say, which for simplicity is assumed to be...

It is an open problem whether an infinite-dimensional amenable Banach algebra exists whose underlying Banach space is reflexive. We give sufficient conditions for a reflexive, amenable Banach algebra to be finite-dimensional (and thus a finite direct sum of full matrix algebras). If A is a reflexive, amenable Banach algebra such that for each maximal left ideal L of A (i) the quotient A/L has t...

In this paper, we describe the primitive ideal space of the $C^*$-algebra $C^*(mathcal G)$  associated to the ultragraph $mathcal{G}$. We investigate the structure of the closed ideals of the quotient ultragraph $  C^* $-algebra  $C^*left(mathcal G/(H,S)right)$ which contain no nonzero set projections and then we characterize all non gauge-invariant primitive ideals. Our results generalize the ...

In this paper we derive a new algorithm for constructing a unitary decomposition of a sequence of matrices in product or quotient form. The unitary decomposition requires only unitary left and right transformations on the individual matrices and amounts to computing the generalized singular value decomposition of the sequence. The proposed algorithm is related to the classical Golub–Kahan proce...

Based on the general operation ◦ of words, called bw-operation, the notions of ◦-primitive words, ◦-closed languages, ◦-bases of languages and operationleft-quotient-closed languages are defined and investigated. These notions turn out to be generalizations of the classical notions of primitive words, plus-closed (star-closed) languages, minimal generating sets and deletion-closed languages. Pr...

0 //K ⊂ //E q //Q //1. Here K stands for kernel and Q stands for quotient. She assumed that K is abelian (= commutative), so she wrote its product as + with identity element 0. She did not assume that Q is abelian, so she wrote its product as · with identity element 1. That explains the joke notation with 0 at the left and 1 at the right. The sequence is exact, which means that K is a normal su...

We survey recent results concerning the complexity of regular languages represented by their minimal deterministic finite automata. In addition to the quotient complexity of the language – which is the number of its (left) quotients, and is the same as its state complexity – we also consider the size of its syntactic semigroup and the quotient complexity of its atoms – basic components of every...

The quotient complexity of a regular language L, which is the same as its state complexity, is the number of left quotients of L. An atom of a non-empty regular language L with n quotients is a non-empty intersection of the n quotients, which can be uncomplemented or complemented. An NFA is atomic if the right language of every state is a union of atoms. We characterize all reduced atomic NFAs ...