(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).

It's silly to reference a book without naming the book, especially with an author with lots of books like Lam. — rschwieb, Aug 07 '17 at 22:10

score 1 · Accepted Answer · answered Aug 07 '17 at 21:38

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.

answered Aug 07 '17 at 21:38

rschwieb

153,510

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

1 Answers1