Commutativity and Ideals in Strongly Graded Rings

In some recent papers by the first two authors it was shown that for any algebraic crossed product A, where A(0), the subring in the degree zero component of the grading, is a commutative ring, each non-zero two-sided ideal in A has a non-zero intersection with the commutant C-A(A(0)) of A(0) in A. This result has also been generalized to crystalline graded rings; a more general class of graded rings to which algebraic crossed products belong. In this paper we generalize this result in another direction, namely to strongly graded rings (in some literature referred to as generalized crossed products) where the subring A(0), the degree zero component of the grading, is a commutative ring. We also give a description of the intersection between arbitrary ideals and commutants to bigger subrings than A(0), and this is done both for strongly graded rings and crystalline graded rings