Factorization theory: From commutative to noncommutative settings

Abstract

Abstract. 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 ω-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 × n upper triangular matrices over a commutative domain using a weak transfer homomorphism to a commutative semigroup.

Factorization theory in commutative monoids

“…Main examples of transfer Krull monoids stem from non-commutative ring theory whence they are beyond the scope of this article. Nevertheless, we mention one example and refer the interested reader to [10,11,86,87] Our next theorem shows that, for the class of finitely generated Krull monoids, the finiteness result for the set of distances and for the set of catenary degrees as well as the structural result for sets of lengths (given in Theorems 3.1 and 5.5) are best possible. 3.…”

On the factorization of non-commutative polynomials (in free associative algebras)

“…For detailed algebraic discussions a lot of definitions (and notations) are necessary. Therefore we formulate most as a special case and refer to [CR94,CR99] for linear representations and [BS15] for the factorization for further information and literature. The factorization in free associative algebras is a natural generalization of that in the (ring of) integers Z.…”

The Catenary Degree of Krull Monoids Ii

“…However, for the Krull monoids under consideration the main result in [18] states that (H) = c(H) holds under a certain mild assumption on the Davenport constant. Our starting point is the following Theorem A (the first statement follows from [19,Theorem 6.4.7], and the characterization of c(H) ∈ [3,4] is given in [18,Corollary 5.6]). …”

“…Then none of the two special cases holds true. By [17,Lemma 3.6], U V has a zero-sum subsequence W 1 ∈ A(G) of length |W 1 | ∈ [2,4], and suppose that |W 1 | is maximal. Then there is a factorization…”