Divisibility Theory of Commutative Rings and Ideal Distributivity

2019/01/18 by Phạm Ngọc Ánh, Anh, P. N., Keith A. Kearnes +3
Mathematics · #Rings, Modules, and Algebras #Advanced Topics in Algebra #Algebraic structures and combinatorial models

paper · pdf · doi:10.48550/arxiv.1901.06304

Abstract

We begin by investigating the class of commutative unital rings in which no two distinct elements divide the same elements. We prove that this class forms a finitely axiomatizable, relatively ideal distributive quasivariety, and it equals the quasivariety generated by the class of integral domains with trivial unit group. We end the paper by proving a representation theorem that provides more evidence to the conjecture that Bézout monoids describe exactly the monoids of finitely generated ideals of commutative unital rings with distributive ideal lattice.

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