Divisibility Theory of Arithmetical Rings with One Minimal Prime Ideal

Abstract

Continuing the study of divisibility theory of arithmetical rings started in [1] and [2], we show that the divisibility theory of arithmetical rings with one minimal prime ideal is axiomatizable as Bezout monoids with one minimal m-prime filter. In particular, every Bezout monoid with one minimal m-prime filter is order-isomorphic to the partially ordered monoid with respect to inverse inclusion, of principal ideals in a Bezout ring with a smallest prime ideal. Although this result can be considered as a satisfactory answer to the divisibility theory of both semihereditary domains and valuation rings, the general representation theory of Bezout monoids is still open.

Key Words:

• Bezout rings
• m-Prime filter
• Spectrum

2010 Mathematics Subject Classification:

• Primary: 06F05
• Secondary: 13A05
• 13F05
• 20M14