Gelfand and exchange rings : their spectra in pointfree topology

Abstract EN

This paper deals with Gelfand and exchange rings A in terms of their spectra, the latter understood not in the usual sense as the spaces of prime, or maximal, ideals, but as their pointfree counterparts, the frames RldA of radical ideals and JRIdA of Jacobson radical ideals.

This approach permits a treatment which is independent of any choice principles but which still leads to the same kind of results as those obtained in the classical choice-dependent setting.

Further, the present results are more general; they imply the classical ones.

In particular, Gelfand rings A are characterized by the normality of RldA, and we show that: a ring A is an exchange ring iff it is Gelfand with zerodimensional JRldA\ a ring is Gelfand iff it is the ring of global elements of what we call a well-supported local ring in the topos of sheaves on a compact regular frame; and a ring is an exchange ring iff it is the ring of global elements of a local ring in the topos of sheaves on a compact zerodimensional frame.

In addition, we provide an analysis of the choice principles required in some previous work in this area.