NONUNITAL PRIME RINGS GRADED BY ORDERED GROUPS

Source abstract

Abstract 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 S e $S_e$ upper S Subscript e . 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 S e $S_e$ upper S Subscript e is G -prime. These results generalise classical theorems by Năstăsescu and Van Oystaeyen to a nonunital setting. As applications, we provide a new and more direct 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.