Proposition 7

The radical of A is the trace on A of the radical of $\widetilde{A}$; it is also equal to the radical of the opposite algebra $A^o($that is, to the intersection of the regular maximal right ideals of A). If the ring $k$ is without radical, then the radical of A is equal to that of $\widetilde{A}$.

The equality $\Re (\widetilde{A})\cap A =\Re (A)$ follows from Proposition 4 of VIII, p. 438, b). Since $\widetilde{A}$ and $\widetilde{A}^o$ have the same radical (VIII, p. 156, Corollary 1), the equality $\Re (A) =\Re (A^o)$ follows. If $k$ is without radical, then the intersection of the maximal left ideals of $\widetilde{A}$ containing A is equal to A. Consequently, $\Re (\widetilde{A})$ is contained in A and therefore equal to $\Re (A)$.