p-spaces and artin-rees property

in this article, we study the artin-rees property in  c(x), in the  rings of fractions of  c(x) and in the factor rings of c(x) . we show that c(x)/(f) is an artin-rees ring if and only if  z(f)  is an open p-space. a necessary and sufficient condition for the local rings of  c(x)   to be artin-rees rings is that each prime ideal in  c(x)  becomes minimal and it turns out that every local ring of c(x)  is an artin-rees ring if and only if  x  is a p-space. finally we have shown that whenever xz(f)  is dense  c-embedded in  x , then  c(x)f  is regular if and only if  xz(f) is a p-space.