# Ring epimorphisms and C(X)

## Michael Barr, W.D. Burgess and R. Raphael

This paper studies the homomorphism of rings of continuous functions $\rho : C(X)\to C(Y)$, $Y$ a subspace of a Tychonoff space $X$, induced by restriction. We ask when $\rho$ is an epimorphism in the categorical sense. There are several appropriate categories: we look at CR, all commutative rings, and R/N, all reduced commutative rings. When $X$ is first countable and perfectly normal (e.g., a metric space), $\rho$ is a CR -epimorphism if and only if it is a R/N-epimorphism if and only if $Y$ is locally closed in $X$. It is also shown that the restriction of $\rho$ to $C^*(X)\to C^*(Y)$, when $X$ is normal, is a CR-epimorphism if and only if it is a surjection.

In general spaces the picture is more complicated, as is shown by various examples. Information about $Spec \rho$ and $Spec \rho$ restricted to the proconstructible set of prime z-ideals is given.

Keywords: epimorphism, ring of continuous functions, category of rings

2000 MSC: 18A20, 54C45, 54B30

Theory and Applications of Categories , Vol. 11, 2003, No. 12, pp 283-308.

http://www.tac.mta.ca/tac/volumes/11/12/11-12.dvi
http://www.tac.mta.ca/tac/volumes/11/12/11-12.ps
http://www.tac.mta.ca/tac/volumes/11/12/11-12.pdf
ftp://ftp.tac.mta.ca/pub/tac/html/volumes/11/12/11-12.dvi
ftp://ftp.tac.mta.ca/pub/tac/html/volumes/11/12/11-12.ps

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