On strongly primary monoids and domains

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 $(R:\hat{R})$ vanishes, then $\Lambda \left(R\right)$ is finite; that is, there exists a positive integer k such that each nonzero nonunit 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 $\Lambda \left(R\right)=\infty .$ In particular, we answer Problem 38 of the open problem list by Cahen et al. in the affirmative. Many of our results are formulated for monoids.

Keywords: 13A05; 13F05; 20M13; Local tameness; one-dimensional local domains; primary monoids; sets of distances; sets of lengths.