Abstract
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 r R 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.
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