Corollary 2

An element of A is invertible if and only if its canonical image in the ring $A/\mathfrak{R}(A)$ is invertible.

The condition is obviously necessary. Let us prove that it is sufficient. Let $x$ be an element of A whose canonical image in the ring $A/\mathfrak{R}(A)$ is invertible. There then exists an element $y$ of A such that $xy$ belongs to $1 +\mathfrak{R}(A)$. By Theorem 1$,xy$ is invertible; hence $x$ is right invertible. The proof that $x$ is left invertible is analogous.