#
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

TAC Home