Using a general definition of a regularity for rings,F- andF- qausi-ideals of a ring are defined. These concepts are shown to be generalizations of ideals or one-sided ideals of a ring. An F-semi primeF—(F-quasi-) ideal of a ringRis also defined.F-regular rings are characterized in terms of F-semi primeF- (F-quasi-) ideals for a large class of polynomial regularities including some well known regularities. A more general characterization of the prime radical β(R) of a ring are given in terms ofF—(F-quasi-) ideals.