pm-Rings and the Prime Ideal Theorem

A commutative ring A with unit is called a pm-ring if every prime ideal of A is contained in a unique maximal ideal, and a Gelfand ring if a + b = 1 in A implies that ( 1 + a r ) ( 1 + b s ) = 0 for some r , s ∈ A . It was shown earlier, in a somewhat circuitous way involving pointfree topology, that “pm implies Gelfand” iff the Prime Ideal Theorem holds. The present note provides an alternative, more direct and entirely ring theoretical proof of a somewhat augmented version of this result.