Contracting the Socle in Rings of Continuous Functions

Abstract

Not every ring homomorphism contracts the socle of its codomain to an ideal contained in the socle of its domain. Rarer still does a homomorphism contract the socle to the socle. We find conditions on a frame homomorphism that ensure that the induced ring homomorphism contracts the socle of the codomain to an ideal contained in the socle of the domain. The surjective among these frame homomorphisms induce ring homomorphisms that contract the socle to the socle. These homomorphisms characterize $P$-frames as those $L$ for which every frame homomorphism with $L$ as domain is of this kind.