Abstract

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=⊕σ∈GRσ is a strongly G-graded ring and H⊲G, then the embedding i:R(H)↪R, where R(H)=⊕σ∈HRσ, is a Zariski extension if and only if H controls the filter ℒ(R−P) 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.