Uniquely Clean ring $A$ with $J(A)=Soc(_AA)$?

Question

A ring $R$ is named uniquely clean if each $r\in R$ has a unique representation $r=u+i$, where $u$ is a unit in $R$ and $i$ is an idempotent of $R$. Now, let $A=\left\{\begin{pmatrix} x & y \\ 0 & x \end{pmatrix}\mid x,y\in \mathbb Z_2\right\}$. I have two issues.

(1) Is this ring uniquely clean?

(2) Is the Jacobson radical of $A$ equal to the left socle of $A$?

Thanks for any help!

Answer 1

After expanding the ring $A$ into its elements, one sees that $A\simeq \mathbb Z_4$. So, the answer to both questions is "yes".