# Divisible By Arbitrary Powers? Not Noetherian

I’ve spent countless hours thinking about associative rings. Yet, during my research today I read a paper by C.U. Jensen  and came across this elementary fact that I never thought about: if $R$ is an integral domain and $x,y$ are nonunits with $y$ divisible by every positive power of $x$ then $R$ is not Noetherian. Perhaps I’ve used this unconsciously, but I had to take a second to prove it.

Here’s the proof: if $y = z_1x = z_2x^2 = \cdots$ then the ideal $(z_1,z_2,\cdots)$ cannot be finitely generated, because $z_n = r_1z_1 + \cdots + r_{n-1}z_{n-1}$ and $y = z_ix^i$ for all $i$ implies that $x$ is actually invertible.

This fact is curious, because it plays into constructing some nonstandard models of the integers as I learned from Jensen’s paper: If $\Ucl$ is a nonprincipal ultrafilter on the natural numbers $\N$, then the product $\Z^\N/\Ucl$ is one such nonstandard model: it satisfies precisely the same first-order sentences that $\Z$ does. Yet, it’s not Noetherian, because the element represented by $(2,4,8,16,32,64,…)$ is not a unit and divisible by every power of $2$.

It’s amazing what basic facts you can still come across in this era. Perhaps someone should write the book 100 Quick Facts You Didn’t Know About Rings?!

 Jensen. Peano rings of arbitrary global dimension. Journal of the London Mathematical Society, 1980, 2, 39-44