Divisibility Theory of Arithmetical Rings with One Minimal Prime Ideal

Anh, Pham Ngoc and Siddoway, M.F. (2016) Divisibility Theory of Arithmetical Rings with One Minimal Prime Ideal. COMMUNICATIONS IN ALGEBRA, 44 (2). pp. 823-836. ISSN 0092-7872

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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. © 2016, Copyright © Taylor & Francis Group, LLC.

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