Factorization theory: From commutative to noncommutative settings

We study the non-uniqueness of factorizations of non zero-divisors into atoms (irreducibles) in noncommutative rings. To do so, we extend concepts from the commutative theory of non-unique factorizations to a noncommutative setting. Several notions of factorizations as well as distances between them are introduced. In addition, arithmetical invariants characterizing the non-uniqueness of factorizations such as the catenary degree, the $\omega$-invariant, and the tame degree, are extended from commutative to noncommutative settings. We introduce the concept of a cancellative semigroup being permutably factorial, and characterize this property by means of corresponding catenary and tame degrees. Also, we give necessary and sufficient conditions for there to be a weak transfer homomorphism from a cancellative semigroup to its reduced abelianization. Applying the abstract machinery we develop, we determine various catenary degrees for classical maximal orders in central simple algebras over global fields by using a natural transfer homomorphism to a monoid of zero-sum sequences over a ray class group. We also determine catenary degrees and the permutable tame degree for the semigroup of non zero-divisors of the ring of $n \times n$ upper triangular matrices over a commutative domain using a weak transfer homomorphism to a commutative semigroup.Comment: 45 page

Order Preservation in Limit Algebras

The matrix units of a digraph algebra, A, induce a relation, known as the diagonal order, on the projections in a masa in the algebra. Normalizing partial isometries in A act on these projections by conjugation; they are said to be order preserving when they respect the diagonal order. Order preserving embeddings, in turn, are those embeddings which carry order preserving normalizers to order preserving normalizers. This paper studies operator algebras which are direct limits of finite dimensional algebras with order preserving embeddings. We give a complete classification of direct limits of full triangular matrix algebras with order preserving embeddings. We also investigate the problem of characterizing algebras with order preserving embeddings.Comment: 43 pages, AMS-TEX v2.