# Characterization of pretty rings

Today I answered this question Characteristics of a pretty ring and wondered whether one could characterize these rings.

Definition: A pretty ring $$R$$ is a ring with unity 1, not a field, and each nonzero element can be written uniquely as a sum of a unit and a nonunit element of $$R$$.

Note that a pretty ring has exactly one unit. Indeed, if $$0 \neq u$$ is not a unit and $$e$$ is a unit, then $$(u+e) + 0 = e + u$$ tells us that $$u+e$$ is not a unit (otherwise we get the contradiction $$u=0$$). Nowe take two units $$e, \tilde{e}$$ then $$e + (u+ \tilde{e}) = \tilde{e} + (u+ e)$$ implies $$e=\tilde{e}$$ and hence $$1$$ is the only unit.

On the other hand every ring (different from $$\mathbb{Z}/ 2 \mathbb{Z})$$ with only one unit is a pretty ring as we can write every $$x\neq 0$$ as $$x = 1 + (x-1).$$

Thus, we have for a unital ring $$R\neq \mathbb{Z}/2 \mathbb{Z}$$: $$R \text{ is a pretty ring} \quad \Leftrightarrow \quad \vert R^\times \vert =1$$ Now, my question is the following:

Are all pretty rings of the form $$R_I = \Pi_{i\in I} (\mathbb{Z}/2\mathbb{Z})$$ where $$I$$ is a set with cardinality different from $$1$$?

The answer is no : indeed the notion of pretty ring can be axiomatized as a first order theory in the language of rings, and it has an infinite model, therefore it has a model of cardinality $$\aleph_0$$, whereas your examples are finite or uncountable.

If you don't know any model theory, that's not an issue : just note that since the only unit is $$1$$, any subring of a pretty ring is itself pretty, and surely an infinite ring has a countable subring.

It is however true that any pretty ring has characteristic $$2$$ : if $$2\neq 0$$, write $$2$$ as $$1+$$ a nonunit to get a contradiction.

• Honestly I don't see how one gets a countable infinite subring. Do you have a reference where I could look up the model theory I'd need to understand your statement? – Severin Schraven Jun 16 at 12:42
• @SeverinSchraven: That is the downward Löwenheim-Skolem theorem. – Henning Makholm Jun 16 at 12:44
• Just look at a subring generated by countably many elements : it is countable – Max Jun 16 at 12:45
• @HenningMakholm Thanks for the key word – Severin Schraven Jun 16 at 12:49
• @Max Of course, that was a stupid question. – Severin Schraven Jun 16 at 12:50

$$\mathbb F_2[X]$$ is pretty but does not have the form you conjecture.

• Indeed, you are right. Nice example! – Severin Schraven Jun 16 at 16:13