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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