# Answer to Question #23461 in Abstract Algebra for jeremy

Question #23461

For R be a subring of a right noetherian ring Q=RS^-1 with a set S ⊆ R ∩ U(Q). Show that: if Q is semiprime, then so is R.

Expert's answer

Let

*a**∈**R*be such that*aRa*= 0. Given ring hasnext property: if*A*is an ideal of*R*, then*AQ*is an idealof*Q.*Applying this to the ideal*A*=*RaR*in*R*, wehave*aQa**⊆**aQ ·*(*RaR*)*Q**⊆**a*(*RaR*)*Q*= 0*,*so*a*=0, and hence R is semiprime ring.
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