Answer to Question #17268 in Abstract Algebra for Tsit Lam

Question

Define the Jacobson radical of R by rad R = {a ∈ R : Ra is left quasi-regular}.
Show that, if R has an identity, the definition of rad R here agrees with classical one .

Answer 1

Assume that 1 ∈ R. The radical rad R = {a ∈ R : Ra is left quasi-regular} can be described as
{a ∈ R : 1 − ra is left-invertible for any r∈ R}.
This is precisely the Jacobson radical for the ring R with identity.