# Answer to Question #23460 in Abstract Algebra for jeremy

Question #23460

Let R be a subring of a right noetherian ring Q with a set S ⊆ R ∩ U(Q) such that every element q ∈ Q has the form rs^−1 for some r ∈ R and s ∈ S. Show that: if Q is prime, then so is R.

Expert's answer

Let

*a*,*b**∈**R*be such that*aRb*= 0. Given ring has nextproperty: if*B*is an ideal of*R*, then*BQ*is an ideal of*Q.*Applying this to the ideal*B*=*RbR*in*R*, we have*aQb**⊆**aQ ·*(*RbR*)*Q**⊆**a*(*RbR*)*Q*= 0*,*so*a*=0 or*b*= 0, and hence R is prime ring.
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