  In this article we generalize a classical result by Passman on primeness of
unital strongly group graded rings to the class of nearly epsilon-strongly
group graded rings which are not necessarily unital. Using this result, we
obtain (i) a characterization of prime $s$-unital strongly group graded rings,
and, in particular, of infinite matrix rings and of group rings over $s$-unital
rings, thereby generalizing a well-known result by Connell; (ii)
characterizations of prime $s$-unital partial skew group rings and of prime
unital partial crossed products; (iii) a generalization of the well-known
characterizations of prime Leavitt path algebras, by Larki and by
Abrams-Bell-Rangaswamy.
