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In this paper we introduce and study a class of ideals between  z -ideals and  z °-ideals (= d -ideals) namely  z r-ideals. A  z r-ideal is a  z -ideal which is at the same time an  r -ideal (an ideal  I  in a ring  R  is called an  r -ideal if for each non-zerodivisor  r  ∈  R  and each  a  ∈  R ,  ra  ∈  I  implies  a  ∈  I ). In contrast to the sum of  z -ideals in  C ( X ) which is a  z -ideal, the sum of  z r-ideals need not be a  z r-ideal. We prove that the sum of every two  z r-ideals of  C ( X ) is a  z r-ideal if and only if  X  is a quasi  F  -space. In  C ( X ) every  z° -ideal is a  z r-ideal and we characterize the spaces  X  for which the converse is also true. We observe that  X  is a cozero complemented space if and only if every (prime)  r -ideal in  C ( X ) is a  z -ideal and whenever every (prime)  z -ideal of  C ( X ) is an  r -ideal it is equivalent to  X  being an almost  P -space. Using these facts it turns out that the set of all  r -ideals and the set of all  z -ideals of  C ( X ) coincide if and only if  X  is a  P -space.

On z r -ideals of C ( X )