Rings in which every prime ideal is contained in a unique maximal right ideal

Our rings have identities and a pm ring is one having the property of the title. In an earlier paper (J. Pure Appl. Algebra 76 (2) (1991) 179-192), we discussed certain conditions on two-sided ideals, equivalent in the commutative case to the pm property. In the present paper we analyze similar conditions on one-sided ideals. As a consequence, we shall prove that, if the Prime Ideal Theorem (PIT) holds, then the right-ideal lattice Id_rR of a ring R is normal if and only if R is Gelfand and satisfies MIT (the Maximal Ideal Theorem). This allows us to redefine Gelfand rings and to establish (in a later paper) Gelfand-Mulvey duality only using PIT rather than MIT.