Divisibility theory in commutative rings: Bezout monoids
Phạm Ngọc Ánh, L. Márki, Péter Vámos · Transactions of the American Mathematical Society · 2012
A ubiquitous class of lattice ordered semigroups introduced by Bosbach in 1991, which we will 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 semihereditary 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 $a,b\in S$, $a\leq b\in S\Longleftrightarrow bS\subseteq aS$), $S$ is a distributive lattice, multiplication is distributive over both meets and joins, and for any $x, y\in S$, if $d=x\wedge y$ and $dx_1=x$, then there is a $y_1\in S$ with $dy_1=y$ and $x_1\wedge y_1=1$. In the present paper, Bezout monoids are investigated by using filters and $m$-prime filters. We also prove analogues of the Pierce and the Grothendieck sheaf representations of rings for Bezout monoids. The question as to whether Bezout monoids describe divisibility in Bezout rings (rings whose finitely generated ideals are principal) is still open.