# Answer to Question #23459 in Abstract Algebra for jeremy

Question #23459

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.

Expert's answer

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.*
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