Remark

Let $x$ and $y$ be elements of A; we say that $x$ is a left adverse of $y$, or that $y$ is a right adverse of $x$, if, in $\widetilde{A},x-e$ is a left inverse of $y-e$, that is, if we have $x+y=xy$. By Proposition 7 and Jacobson’s theorem (VIII, p. 156, Theorem 1), the radical of A consists of the elements $x$ of A such that $ux-e$ is left invertible in $\widetilde{A}$ for every $u$ in $\widetilde{A}$. Since we have $(a+\lambda e)x-e=ax+\lambda x-e$ (for $a\in A,\lambda \in k$), the radical of A is the set of elements $x$ of A such that $ax+\lambda x$ has a left adverse in A for every $a\in A$ and $\lambda \in k$.