Suppose $R$ a ring with 1 and let $U (R)$ denote its invertible elements. If $(M = R \setminus U(R), +)$ is a group, show that $M$ is a left ideal of $R$.
I know that $U (R) M \subseteq M$. But I don't know how to prove $M M \subseteq M$. Can anyone help?
Take $r\in R$ and $x\in M$ and suppose that $rx\in U(R)$. Then, there is $u\in R$ such that $rxu=urx=1$. So the element $e=xur$ is idempotent, since $ee=xurxur=xur=e$, we have $e\neq 1$ because $x$ is not invertible and $e\neq 0$ because $R\neq\{0\}$. So $e$ and $1-e$ are non-invertible idempotents, that is $e,1-e\in M$ and their sum is $1\notin M$, contradicting the fact that $(M,+)$ is a group.