  We show that if a groupoid graded ring has a certain nonzero ideal property
and the principal component of the ring is commutative, then the intersection
of a nonzero twosided ideal of the ring with the commutant of the principal
component of the ring is nonzero. Furthermore, we show that for a skew groupoid
ring with commutative principal component, the principal component is maximal
commutative if and only if it is intersected nontrivially by each nonzero ideal
of the skew groupoid ring. We also determine the center of strongly groupoid
graded rings in terms of an action on the ring induced by the grading. In the
end of the article, we show that, given a finite groupoid $G$, which has a
nonidentity morphism, there is a ring, strongly graded by $G$, which is not a
crossed product over $G$.
