Answer to Question #23461 in Abstract Algebra for jeremy

Question

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.

Answer 1

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.