On the sheaf characterization of Gelfand rings

Abstract

This paper deals with the commutative rings with unit in which a+b = 1 implies that (1+ar)(1+bs) = 0 for suitable r and s. These rings have been considered, alternatively, in terms of conditions on their maximal ideals (DeMarco-Orsatti 1971, Mulvey 1979, Johnstone 1982) which are equivalent to the above in the presence of the Axiom of Choice (Banaschewski 2000).