(Lam Prop 5.6) R $\subseteq S$, then under what condition rad(S) $\cap$ R $\subseteq$ rad (R)?

Question

The condition given in the text is $_{R}R$ direct summand of $_{R}S$.

But I think I don't need this condition because if x $\in$ rad(S) $\cap$ R, then for all y $\in$ S, 1-xy is left invertible (property of rad(S)), but since R $\subseteq$ S, this is equally true for y $\in$ R and hence x $\in$ rad(R).

Answer 1

$1-xy$ is right invertible by an element of $S$, but why should that element happen to be in $R$? Maybe there aren't any right inverses in$R$ after all!

That is the gap in the reasoning you gave.