\(pm\)-rings and the prime ideal theorem
A commutative ring \(A\) with unit is a \textit{\(pm\)-ring} if every prime ideal of \(A\) is contained in a unique maximal ideal. These rings, originally introduced by \textit{G. De Marco} and \textit{A. Orsatti} [Proc. Am. Math. Soc. 30, 459--466 (1971; Zbl 0207.05001)], constitute a natural class of abstract rings that capture many features of the concrete rings of continuous real-valued functions on topological spaces. On the other hand, a \textit{Gelfand ring} is a commutative ring with unit in which \(a+b=1\) implies that \((1+ar)(1+bs)=0\) for some \(r,s\). In an earlier paper [Arab. J. Sci. Eng., Sect. C, Theme Issues 25, No. 2, 3--22 (2000; Zbl 1271.13052)], the author of the paper under review had shown by methods of pointfree topology that the Prime Ideal Theorem (PIT) is equivalent to the statement that every \(pm\)-ring is a Gelfand ring, whereas the converse holds without any choice principles. In the present note he provides direct, purely ring-theoretical arguments for these facts. The clue is the observation that an idempotent-generated ring with torsion-free additive group cannot be Gelfand and must therefore contain a prime ideal (without appealing to the PIT); and that any Boolean algebra can be embedded into the Boolean algebra of idempotents of a commutative unitary ring with torsion-free additive group. By similar methods, it is shown that the \textit{weakly Gelfand rings}, defined by the property that for maximal ideals \(P \neq Q\) there exist \(r \notin P\) and \(s \notin Q\) such that \(rs = 0\), are exactly the same as the Gelfand rings if and only if the Axiom of Choice holds.
scientific article; zbMATH DE number 5969484
scientific article