Factorizations in bounded hereditary Noetherian prime rings

Abstract

If $H$ is a monoid and $a=u_1 \cdots u_k \in H$ with atoms (irreducible elements) $u_1, \ldots, u_k$, then $k$ is a length of $a$, the set of lengths of $a$ is denoted by $\mathsf L(a)$, and $\mathcal L(H)=\{\,\mathsf L (a) \mid a \in H \,\}$ 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 $R^\bullet$ 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 $R^{\bullet}$ to the monoid $B$ of zero-sum sequences over a subset $G_{\textrm{max}}(R)$ of the ideal class group $G(R)$. This implies that the systems of sets of lengths, together with further arithmetical invariants, of the monoids $R^{\bullet}$ 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.