35 research outputs found
Factorizations of Elements in Noncommutative Rings: A Survey
We survey results on factorizations of non zero-divisors into atoms
(irreducible elements) in noncommutative rings. The point of view in this
survey is motivated by the commutative theory of non-unique factorizations.
Topics covered include unique factorization up to order and similarity, 2-firs,
and modular LCM domains, as well as UFRs and UFDs in the sense of Chatters and
Jordan and generalizations thereof. We recall arithmetical invariants for the
study of non-unique factorizations, and give transfer results for arithmetical
invariants in matrix rings, rings of triangular matrices, and classical maximal
orders as well as classical hereditary orders in central simple algebras over
global fields.Comment: 50 pages, comments welcom
Matrix-isomorphic maximal Z-orders
AbstractWe construct many pairwise non-isomorphic maximalZ-ordersAandBwhich have isomorphicnbynmatrix rings for every positive integern≠1. In most casesAalso has the property that every one-sided ideal ofM2(A) is principal but not every one-sided ideal ofAis principal
Idealiser rings of injective dimension one
We study a certain type of prime Noetherian idealiser ring R of injective dimension 1, and prove for instance that the idempotent ideals of R are projective and that every non-zero projective ideal of R is uniquely of the form UE for some invertible ideal U and idempotent ideal E of R. Formulae are given for the number of idempotent ideals of R and the number of orders which contain R. </jats:p