When is $\mathbb{G}_m(R)$ enough to determine $R$? - MathOverflow

When is $\mathbb{G}_m(R)$ enough to determine $R$?

2011-11-01T00:04:57Z

Say I have a ring, $R$, with 1 which I consider my universe, and I know its group of units $G=\mathbb{G}_m(R)$. Then given a subgroup, $H\le G$, can I determine if there is there a subring $S_H$ such that $\mathbb{G}_m(S)=H$? If so, is $S_H$ unique with this group of units? If so, is there in fact a--canonical in the sense above--1-1 correspondence between subgroups of $G$ and subrings of $R$ with 1? Preliminary attempts at a solution don't indicate any problems with the truth of the statement, but naturally one should be skeptical of limited data especially in a subject with so many intricacies as groups and rings.

The motivating example is $\overline{\mathbb{Q}}/\mathbb{Q}$, due to some interesting number theory that could come out of such a correspondence.

In the case of fields the question is supposed to collapse into the question "Can I add 0 to a subgroup of the group of units of some big field and get a subfield without doing anything else?"

There is no possibility for general rings, but are there assumptions on $R$ or $G$ which can ensure existence or uniqueness? And it is also fine to induce assumptions on what kind of $S$ we are allowed to have as well, fields instead of just rings for example.

Answer by Steven Landsburg for When is $\mathbb{G}_m(R)$ enough to determine $R$?

2011-11-01T00:26:33Z

Uniqueness is hopeless; let $k$ be any reduced ring and $R=k[x]$. Then $k\subset R$ and $R\subset R$ have the same group of units.

Existence is also hopeless in general: Let $S=Z/5Z$, and let $H$ be the subgroup conisting of 1 and 4.

Answer by Salvatore Siciliano for When is $\mathbb{G}_m(R)$ enough to determine $R$?

2011-11-01T11:29:19Z

It is also worth to recall that if $F$ is a field then a free algebra $F[X]$ has $F^\times$ as group of units for any set $X$...

Anyway, there is an extensive literature devoted to study rings with a fixed group if units. A sample of this is the paper:

I. Stewart: Finite rings with a specified group of units, Math. Z. 126 (1972), 51-58.