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Answer to Question #17268 in Abstract Algebra for Tsit Lam

Question #17268
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 .
Expert's answer
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.

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