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) $u_1,\dots ,u_k$, 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 $G_{\text{max}}(R)$ 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.

50 pages

References in corpus (3)

• Quantum cluster algebras and quantum nilpotent algebras
• Factorizations of Elements in Noncommutative Rings: A Survey
• Non-commutative Krull monoids: A divisor theoretic approach and their arithmetic

Cited by in corpus (12)

• Systems of sets of lengths: Transfer Krull monoids versus weakly Krull monoids
• Factorizations of Elements in Noncommutative Rings: A Survey
• Sets of Arithmetical Invariants in Transfer Krull Monoids
• On the Arithmetic of Power Monoids and Sumsets in Cyclic Groups
• Long sets of lengths with maximal elasticity
• On half-factoriality of transfer Krull monoids
• Definite orders with locally free cancellation
• A characterization of finite abelian groups via sets of lengths in transfer Krull monoids
• A realization result for systems of sets of lengths
• Factorization in Finitely-Presented Monoids
• On transfer Krull monoids
• Elasticities of Orders in Central Simple Algebras