Examples of local ring with principal group units $U_{1}=U_{1}^{2}$

Question

Let $A$ be a local ring with maximal ideal $M$. if we consider $U_{1}=1+m$.

I am trying to find some examples of local rings where the condition $U_{1}=U_{1}^{2}$ holds.

I was thinking like local rings where you localize at one prime $p$ or discrete valuation rings like the $p-$adic integers but i do not how to prove if the condition $U_{1}=U_{1}^{2}$ is true in these cases.

I would appreciate some help or hint for this.

Thanks!

Answer 1

As you say, take $A=\Bbb Z_p$ the $p$-adics with $p$ odd. Then each $a\in A$ with $a\equiv1\pmod p$ has a square root $b\equiv1\pmod p$. Just Hensel lift $x=1$ to a solution of $x^2-a=0$.

Answer 2

For a very simple example, let $A$ be any field. Then $U_1$ is just $\{1\}$.

Or, a bit more generally, if the maximal ideal $M$ satisfies $M^2=0$, then multiplication on $U_1$ is really just addition on $M$, since $(1+x)(1+y)=1+x+y+xy=1+x+y$ for $x,y\in M$. So, $U_1^2=U_1$ as long as $2$ is invertible in $A$. You can get an example where this happens by starting with any commutative ring with a maximal ideal that does not contain $2$ and then modding out the square of the maximal ideal.