# Answer to Question #17656 in Abstract Algebra for Tsit Lam

Question #17656

Using the definition of rad R as the intersection of the maximal left ideals, show directly that rad R is an ideal.

Expert's answer

For

*y**∈**rad**R*,*r**∈**R*, and many maximal left ideal, we must show that*yr**∈**m. Assume otherwise; then**Rr*+ m =*R*. Consider the left*R*-modulehomomorphism*ϕ*:*R → R/*m definedby*ϕ*(*x*) =*xr (mod m)*.Since*Rr*+ m =*R*,*ϕ*is onto. This implies thatker(*ϕ*) is a maximal left ideal.Therefore,*y**∈**ker(**ϕ*), so we have 0 =*ϕ*(*y*) =*yr (mod m)*, acontradiction.
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