Answer to Question #17267 in Abstract Algebra for Tsit Lam

Question

Show that rad R is a quasi-regular ideal which contains every quasi-regular left (resp. right) ideal of R. (In particular, rad R contains every nil left or right ideal of R.)

Answer 1

To show that rad R is a left ideal, it suffices to check that, if a, b ∈rad R, then r(a + b) is left quasi-regular for any r ∈ R. For any c , d∈ R, note that c ◦ d ◦ r(a + b)= c ◦ [d + ra + rb − dr(a + b)] = c◦ [d ◦ ra + (r − dr)b].
Choose d such that d ◦ ra =0; then choose c such that c ◦ (r − dr)b = 0. The above equation shows that c ◦ d is a left quasi-inverse of r(a+ b), as desired. To show that rad R is also a right ideal,we must check that a ∈rad R and s ∈ R imply as ∈rad R. For any r ∈ R, sra is left quasiregular, so r(as) is also left quasi-regular. This shows that as ∈rad R.
Next, we show that rad R is a quasi-regular ideal.It suffices to show that every element a ∈rad R is left quasi-regular. Since a² ∈ Ra is left quasi-regular, there exists b ∈ R such that b ◦ a² = 0. But then
(b ◦ (−a)) ◦ a =b ◦ ((−a) ◦ a) = b ◦ a² = 0
shows that a is left quasi-regular. By definition, rad R contains every quasi-regular left ideal. Let I be any quasi-regular right ideal. For a ∈ I, aR is right quasi-regular, therefore quasi-regular. We also see that Ra is (left) quasi-regular, so I ⊆rad R, as desired.