On noncommutative bounded factorization domains and prime rings

TL;DR

This paper investigates bounded factorizations in noncommutative rings, providing conditions for when such rings have this property and presenting a counterexample of a noncommutative algebra lacking bounded factorizations.

Contribution

It establishes sufficient conditions for noncommutative noetherian prime rings to have bounded factorizations and constructs a finitely presented algebra that does not.

Findings

Noncommutative noetherian prime rings can have bounded factorizations under certain conditions.

A finitely presented semigroup algebra can be atomic without satisfying ACCP.

Counterexample shows not all noncommutative atomic domains have bounded factorizations.

Abstract

A ring has bounded factorizations if every cancellative nonunit $a \in R$ can be written as a product of atoms and there is a bound $\lambda(a)$ on the lengths of such factorizations. The bounded factorization property is one of the most basic finiteness properties in the study of non-unique factorizations. Every commutative noetherian domain has bounded factorizations, but it is open whether such a result holds in the noncommutative setting. We provide sufficient conditions for a noncommutative noetherian prime ring to have bounded factorizations. Moreover, we construct a (noncommutative) finitely presented semigroup algebra that is an atomic domain but does not satisfy the ascending chain condition on principal right or left ideals (ACCP), whence it does not have bounded factorizations.