Spontaneous atomicity for polynomial rings with zero-divisors

TL;DR

This paper demonstrates that certain rings with zero-divisors can be non-atomic or antimatter, yet their polynomial extensions can be strongly atomic, with controlled factorization lengths.

Contribution

It constructs examples of rings that are antimatter but have strongly atomic polynomial extensions, revealing new insights into atomicity properties in ring theory.

Findings

Existence of antimatter rings with strongly atomic polynomial extensions

Polynomial factorizations have length at most deg(f(t))+2

Non-atomic rings can have atomic polynomial extensions

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.