DIVISIBILITY THEORY IN COMMUTATIVE RINGS:BEZOUT MONOIDS

摘要

A ubiquitous class of lattice ordered semigroups introduced by I3osbach in 1991, which we will call Bezout monoids. seems to be the ap-propriate structure for the study of divisibility in various classical rings like (CCI) domains (including TFD's). rings of low dimension (including semi-hereditary rings), as well as certain subdirect products of such rings and cer-tain factors of such subdirect products. A Bczout monoid is a commutative nionoid S with 0 such that under the natural partial order (for a, b ∈ s. b ∈ s→∪aS).S is a distributive lattice. multiplication is dis-tributive over both meets and joins. and for any x. y∈ S. if d =xΔy and dx_1 = x, then there is a y_1 ∈ S wit dy_1=y and x_1 Δy_1 =1 .In the present paper. liezont nionoids 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.