Divisibility theory in commutative rings: Bezout monoids

A ubiquitous class of lattice ordered semigroups introduced by Bosbach, which we call Bezout monoids, seems to be the appropriate structure for the study of divisibility in various classical rings like GCD domains (including UFD’s), rings of low dimension (including semi-hereditary rings), as well as certain subdirect products of such rings and certain factors of such subdirect products. A Bezout monoid is a commutative monoid S with 0 such that under the natural partial order (for a, b ∈ S, a ≤ b ∈ S ⟺ bS ⊆ aS), S is a distributive lattice, multiplication is distributive over both meets and joins, and for any x, y ∈ S, if d = x ∧ y and dx1 = x then there is a y1 ∈ S with dy1 = y and x1 ∧ y1 = 1. We investigate Bezout monoids by using filters and m-prime filters, and describe all homorphisms between them. We also prove analogues of the Pierce and the Grothendieck sheaf representations of rings for Bezout monoids. The question whether Bezout monoids describe divisibility in Bezout rings (rings whose finitely generated ideals are principal) is still open.

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