We introduce crossed product-like rings, as a natural generalization of crystalline graded rings, and describe their basic properties. Furthermore, we prove that for certain pre-crystalline graded rings and every crystalline graded ring A, for which the base subring A0 is commutative, each non-zero two-sided ideal has a nonzero intersection with CA(A0), i.e. the commutant of A0 in A. We also show that in general this property need not hold for crossed product-like rings.