Answer to Question #23460 in Abstract Algebra for jeremy

Question

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.

Answer 1

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.