Let G be a group with identity element e, and suppose that S is an associative G-graded ring that is not necessarily unital. In the case where G is an ordered group, we show that a graded ideal is prime if and only if it is graded prime. Consequently, in that setting, a graded ring is prime if and only if it is graded prime. For any group G, if S is what we call ideally symmetrically G-graded, then we show that there is a bijective correspondence between the G-graded prime ideals of S and the G-prime ideals of Se. We use this correspondence in the case where G is ordered and S is ideally symmetrically G-graded to show that S is prime if and only if Se is G-prime. These results generalize classical theorems by Năstăsescu and Van Oystaeyen to a nonunital setting. As applications, we provide a new proof of a primeness criterion for Leavitt path rings and establish conditions for primeness of symmetrically G-graded subrings of group rings over fully idempotent rings.