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arXiv · 1901.06304

Divisibility Theory of Commutative Rings and Ideal Distributivity

P. N. Anh·Keith A. Kearnes·Agnes Szendrei
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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 B\'ezout monoids describe exactly the monoids of finitely generated ideals of commutative unital rings with distributive ideal lattice.

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P. N. Anh, Keith A. Kearnes, Agnes Szendrei. 2019-01-18. Divisibility Theory of Commutative Rings and Ideal Distributivity. https://arxiv.org/abs/1901.06304

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