Ring with only one prime element up to associates

Question

Let $R$ be a commutative ring.

Introduce equivalent relation in the set of all prime elements of $R$ (we call this set $P$).

$p_1\sim p_2$ if only if $p_1$ and $p_2$ are associate ($p_1｜p_2$ and $p_2｜p_1$).

Then, I want determine all $R$ which satisfies
(★) $P/\sim$ has only one element.

For example, when $R＝\Bbb{Z}_p$, $P＝\{pu｜u \text{ is a unit element of }R\}$, $P/\sim＝\{[p]\}$, so $R＝\Bbb{Z}_p$ satisfies (★).

I couldn't come up with another ring which satisfies (★), and I wonder this is essential characteristic of ring of $p$ adic integers, like saying ,'$\Bbb{Z}_p$ is a ring with only one prime (element), $p$'.

One way to define a DVR is a UFD with only one irreducible element. — Douglas Molin, Mar 05 '22 at 13:23
Of course, it is only unique up to multiplication by units. I forgot to add that. — Douglas Molin, Mar 05 '22 at 14:10
Zp has infinitely many units, so 'Zp has finitely many prime elements, but they are essentially the same one', my image. — Poitou-Tate, Mar 05 '22 at 14:32

score 2 · Accepted Answer · answered Mar 05 '22 at 17:45

2

Here's an example that is a domain but not a DVR: let $R$ be a valuation ring with value group $\Bbb Z \times \Bbb Z$ with lexicographic order. An element is prime iff it has valuation $(0,1)$ and all those elements are associate.

answered Mar 05 '22 at 17:45

Lukas Heger

25,763

score 1 · Answer 2 · answered Mar 05 '22 at 13:15

1

There are a lot of such rings: every discrete valuation ring has the property (*). For example every localization of the polynomial ring $K[X]$ over a field $K$ with respect to any prime polynomial $p\in K[X]$.

answered Mar 05 '22 at 13:15

Hagen Knaf

9,387

Torsten Schoeneberg · Answer 3 · 2022-03-10T05:31:54.053

As Hagen Knaf's answer says, any DVR (almost by definition) is an example of this. Quotients of DVRs give more examples. Polynomial rings over (e.g.) $\mathbb Z/p^n$ ($n \ge 2$) give more noetherian examples which are not domains.

Lukas Heger's answer gives other examples among domains which are not noetherian.

Another example among noetherian domains was given by user reuns in answer to Noetherian domain with unique principal prime ideal that is not a DVR.

Another example, which is neither a domain nor noetherian, was given here: https://mathoverflow.net/a/155250/27465.

Ring with only one prime element up to associates

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