International Journal of Mathematics and Mathematical Sciences
Volume 22 (1999), Issue 2, Pages 411-415

Control subgroups and birational extensions of graded rings

Salah El Din S. Hussein

Department of Mathematics, Faculty of Science, Ain Shams University, Abbassia, Cairo 11566, Egypt

Received 17 April 1998

Copyright © 1999 Salah El Din S. Hussein. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


In this paper, we establish the relation between the concept of control subgroups and the class of graded birational algebras. Actually, we prove that if R=σGRσ is a strongly G-graded ring and HG, then the embedding i:R(H)R, where R(H)=σHRσ, is a Zariski extension if and only if H controls the filter (RP) for every prime ideal P in an open set of the Zariski topology on R. This enables us to relate certain ideals of R and R(H) up to radical.