Spontaneous atomicity for polynomial rings with zero-divisors
Commutative Algebra
2016-12-20 v1
Abstract
In this paper, we show that it is possible for a commutative ring with identity to be non-atomic (that is, there exist non-zero nonunits that cannot be factored into irreducibles) and yet have a strongly atomic polynomial extension. In particular, we produce a commutative ring with identity, R, that is antimatter (that is, R has no irreducibles whatsoever) such that R[t] is strongly atomic. What is more, given any nonzero nonunit f(t) in R[t] then there is a factorization of f(t) into irreducibles of length no more than deg(f(t)) + 2.
Cite
@article{arxiv.1612.05976,
title = {Spontaneous atomicity for polynomial rings with zero-divisors},
author = {Jim Coykendall and Stacy Trentham},
journal= {arXiv preprint arXiv:1612.05976},
year = {2016}
}