On the Arithmetic of Strongly Primary Monoids

Arithmetic of Mori domains and monoids

“…If all D i are as asserted, then D is a v-noetherian G-monoid with finite class group C( D), and D is locally tame by [26,Theorem 3.5]. Thus H has the asserted properties by Proposition 6.4.…”

“…For general information on local tameness and its relevance in factorization theory we refer to [23]. Recent results can be found in [2,8,12,13,25,26]. The main result of this paper is Theorem 5.…”

“…A second extreme case of v-noetherian monoids we want to describe are primary monoids, that is, monoids which have only a single non-empty prime s-ideal. If the conductor (H : H ) of such a monoid H is empty, then H may fail to be locally tame (examples of such monoids can be found in [26]). Therefore it is natural to claim, in Definition 4.1, that H has non-empty conductor.…”

“…(b) are necessary. In the special case of a primary v-noetherian monoid H with (H : H ) = ∅, the root closure of H is a weakly C-monoid if and only if C( H ) is finite (see[26, Theorem 3.5]). The next proposition sheds light on the relationship between weakly C-monoids on the one hand, and C-monoids, Krull monoids and finitely generated monoids on the other hand.…”

A realization theorem for sets of lengths

“…In locally tame, strongly primary monoids (e.g., in one-dimensional local noetherian domains) the structure of sets of lengths is simpler than it is in Krull monoids (see [10,Theorem 4.3.6]), and a first realization theorem for sets of lengths for this class of monoids was recently given in [12].…”