# 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 .

Show that, if R has an identity, the definition of rad R here agrees with classical one .

Expert's answer

Assume that 1

This is precisely the Jacobson radical for the ring

*∈**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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