Ring theory and pointfree topology.
In this paper the author reviews the `pointfree' version of the sheaf representation theory of rings (using sheaves on locales rather than on classical topological spaces). The advantage of working with locales in this context is that one can eliminate the uses of the Boolean Prime Ideal Theorem, which are required in the classical case to ensure that the rings in question have enough prime ideals (equivalently, that the locales over which they are represented have enough points). As an example, he gives a pointfree (and hence choice-free) proof of an old result of the reviewer, that a ring is an exchange ring if and only if it is a Gelfand ring and its maximal spectrum is zero-dimensional. He also considers the sheaf representation of ordered rings, and shows that an \(l\)-ring is an \(f\)-ring if and only if it has a sheaf representation by a totally ordered ring; once again, the `pointed' version of this result is in the reviewer's book [Stone spaces. Cambridge: Cambridge Univ. Press (1982; Zbl 0499.54001)].
scientific article; zbMATH DE number 2056840
scientific article