DIVISIBILITY THEORY IN COMMUTATIVE RINGS: BEZOUT MONOIDS

作者：Anh P N^*; Marki L; Vamos P

来源：Transactions of the American Mathematical Society, 2012, 364(8): 3967-3992.

DOI：10.1090/S0002-9947-2012-05424-9

• Bezout ring
• divisibility semigroup
• Bezout monoid
• spectrum
• prime filter
• sheaf

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%26apos;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 is an element of S, a %26lt;= b is an element of S double left right arrow bS subset of aS), S is a distributive lattice, multiplication is distributive over both meets and joins, and for any x, y is an element of S, if d = x Lambda y and dx(1) = x, then there is a y(1) is an element of S with dy(1) = y and x(1) Lambda 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.

• 出版日期2012-8