On strongly primary monoids and domains

TL;DR

This paper characterizes strongly primary domains and monoids, establishing finiteness conditions on factorizations and linking local properties to global tameness, thereby solving an open problem in commutative ring theory.

Contribution

It introduces new characterizations of strongly primary domains and monoids, especially relating to factorization finiteness and tameness, and resolves an open problem in the field.

Findings

Domains with vanishing conductor have finite factorization length

Every strongly primary domain is locally tame

A domain is globally tame iff its factorization length set is infinite

Abstract

A commutative integral domain is primary if and only if it is one-dimensional and local. A domain is strongly primary if and only if it is local and each nonzero principal ideal contains a power of the maximal ideal. Hence one-dimensional local Mori domains are strongly primary. We prove among other results, that if $R$ is a domain such that the conductor $(R:\widehat R)$ vanishes, then $\Lambda(R)$ is finite, that is, there exists a positive integer $k$ such that each non-zero non-unit of $R$ is a product of at most $k$ irreducible elements. Using this result we obtain that every strongly primary domain is locally tame, and that a domain $R$ is globally tame if and only if $\Lambda(R)=\infty$. In particular, we answer Problem 38 in {P.-J. Cahen, M.~Fontana, S.~Frisch, and S.~Glaz, Open problems in commutative ring theory, Commutative Algebra, Springer 2014} in the affirmative. Many of our results are formulated for monoids.