Factorizations in Bounded Hereditary Noetherian Prime Rings

Abstract

If H is a monoid and a = u 1 · · · u k ∈ H with atoms (irreducible elements) u 1 , . . . , u k , then k is a length of a, the set of lengths of a is denoted by L(a), and L(H) = { L(a) | a ∈ H } is the system of sets of lengths of H. Let R be a hereditary Noetherian prime (HNP) ring. Then every element of the monoid of non-zero-divisors R • can be written as a product of atoms. We show that, if R is bounded and every stably free right R-ideal is free, then there exists a transfer homomorphism from R • to the monoid B of zero-sum sequences over a subset G max (R) of the ideal class group G(R). This implies that the systems of sets of lengths, together with further arithmetical invariants, of the monoids R • and B coincide. It is well-known that commutative Dedekind domains allow transfer homomorphisms to monoids of zero-sum sequences, and the arithmetic of the latter has been the object of much research. Our approach is based on the structure theory of finitely generated projective modules over HNP rings, as established in the recent monograph by Levy and Robson. We complement our results by giving an example of a non-bounded HNP ring in which every stably free right R-ideal is free but which does not allow a transfer homomorphism to a monoid of zero-sum sequences over any subset of its ideal class group.The principal aim of factorization theory is to describe the various phenomena of non-uniqueness of factorizations by suitable arithmetical invariants (such as sets of lengths), and to study the interaction of these arithmetical invariants with classical algebraic invariants of the underlying ring (such as class groups). Factorization theory has its origins in algebraic number theory and in commutative algebra. The focus has been on commutative Noetherian domains, commutative Krull domains and monoids, their monoids of ideals, and their monoids of modules. We refer to [And97 And97, Nar04 Nar04, Cha05 Cha05, GHK06 GHK06, Ger09 Ger09, BW13 BW13, Fac12 Fac12, LW12 LW12, BW13 BW13, FHL13 FHL13, CFGO16 CFGO16] for recent monographs, conference proceedings, and surveys on the topic.A key strategy, lying at the very heart of factorization theory, is the construction of a transfer homomorphism from a ring or monoid of interest to a simpler one. This allows one to study the arithmetic of the simpler object and then pull back the information to the original object of interest. An exposition of transfer principles in the commutative setting can be found in [GHK06 GHK06, Section 3.2]. The best investigated class of simpler objects occurring in this context is that of monoids of zero-sum sequences over subsets of abelian groups. These monoids are commutative Krull monoids having a combinatorial flavor; questions about factorizations are reduced to questions about zero-sum sequences, which are studied with methods from additive combinatorics. We refer the reader to the survey [Ger09 Ger09] for the interplay between additive combinatorics and the arithmetic of monoids.Until relatively recently, the study of factorizations of elem...

Arithmetical invariants of local quaternion orders

“…In this case stable isomorphism of finitely generated projective R-modules implies isomorphism, and R has trivial ideal class group (see [LR11,§40]). As a consequence of [Sme17,Theorem 4.4], we then have ρ(R • ) = 1, ∆(R • ) = ∅, and c d (R • ) ≤ 2. Moreover, more precise information about unique factorization is available in the form of [Sme17,Proposition 4.12].…”

“…As a consequence of [Sme17,Theorem 4.4], we then have ρ(R • ) = 1, ∆(R • ) = ∅, and c d (R • ) ≤ 2. Moreover, more precise information about unique factorization is available in the form of [Sme17,Proposition 4.12]. (In this case, ρ(R • ) = 1 and ∆(R • ) = ∅ can also be derived using [Est91].…”

“…The determinant is again a transfer homomorphism (see [Est91]). Moreover, R • is composition series factorial, but not similarity factorial (see [Sme17,Proposition 4.12]).…”

“…The elasticity of R • is then ρ(R • ) = sup{ ρ(a) : a ∈ R • } ∈ R ≥1 ∪{∞}. The study of arithmetical invariants, such as sets of lengths and elasticities, is part of factorization theory, a field which has been well-developed in the commutative setting (see [And97,Cha05,GHK06,FHL13,CFGO16,Ger16] for recent monographs and surveys) and which has been recently extended to noncommutative settings (see [BBG14,BS15,Sme16,Sme17,BJ16,BHL17] for recent results).…”

“…This holds in particular when R is a ring of algebraic integers in a number field, where the class group is the usual one. More recently, these results have been extended to a noncommutative setting: If R is a hereditary order in a central simple algebra over a number field, then R • is a transfer Krull monoid as well (see [Sme17] or [Sme13,BS15] for the special case of maximal orders). Here, the class group is isomorphic to a ray class group of the center.…”