Rings in which every prime ideal is contained in a unique maximal right ideal
In order to extend to rings the Gelfand duality between compact spaces and commutative \(C^*\) algebras, \textit{C. J. Mulvey} [J. Algebra 56, 499--505 (1979; Zbl 0498.16035)] introduced the following condition: Let \(R\) be a ring such that for any distinct maximal right ideals \(M\), \(M'\), there exist \(a\notin M\), \(a'\notin M'\) in \(R\) with \(aRa'=0\). Then \(R\) is called a Gelfand ring. It follows that every prime ideal in such a ring is contained in a unique maximal right ideal. In this paper, rings with this last property and related properties of the space \(\text{spec}_ r R\) of all prime right ideals and \(\text{spec }R\) are considered. The second section of the paper is concerned with set theoretical axioms; in particular it is shown that under the assumption of the prime ideal theorem, and not assuming the axiom of choice, the lattice of right ideals of \(R\) is normal if and only if \(R\) satisfies the maximal ideal theorem and is Gelfand.
scientific article; zbMATH DE number 60684
scientific article