On strongly primary monoids and domains

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.