Answer to Question #23459 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 B is an ideal of R, then BQ is an ideal of Q.

Answer 1

For any q ∈ Q, we need to show that qBQ ⊆ BQ. Write q =rs^−1 as above. Theascending sequence of right ideals s^(−1)BQ ⊆ s^(−2)BQ ⊆· · · in the right noetherian ring Q showsthat s^(−n)BQ = s^(−(n+1))BQ forsome n ≥ 1, and thus BQ =s^(−1)BQ. From this we see that qBQ = r(s^(−1)BQ)= rBQ ⊆ BQ.