# Answer to Question #17360 in Algebra for Mohammad

Question #17360

Let R be the ring of all continuous real-valued functions on a topological space A. Show that R is J-semisimple, but “in most cases” not von Neumann regular.

Expert's answer

The following are clearly maximalideals of

To see that in most cases

space

*R*: m*=*_{a}*{f**∈**R*:*f*(*a*) = 0*},*where*a**∈**A*. Therefore, rad*R**⊆(intersection)*m*=*_{a}*{f**∈**R*:*f*(*A*) = 0*}*= 0*.*To see that in most cases

*R*isnot von Neumann regular, consider any nonsingleton connected compact Hausdorffspace

*A*. Then the only idempotents in*R*are 0 and 1. Assume*R*isvon Neumann regular. For any nonzero*f**∈**R*,*fR*=*eR*for some idempotent*e**∈**R*, so we must have*fR*=*R*, i.e.*f**∈**U(**R*). Therefore,*R*is a field. The known classificationtheorem for maximal ideals of*R*then implies that*|A|*= 1, acontradiction.
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