An ideal I in a commutative ring R is called a z^{°}ideal if I consists of zerodivisors and for each a ∈ I the intersection of all minimal prime ideals containing a is contained in I. We
characterize topological spaces X for which zideals and z^{°}ideals
coincide in C(X), or equivalently the sum of every two ideals consisting entirely of zero divisors consists entirely of zero divisors. Basically disconnected spaces, extremally disconnected and Pspaces are
characterized in terms of z^{°}ideals. Finally, we construct two topological almost Pspaces X and Y which are note Pspaces and such that in C(X) every prime z^{°}ideal is either a minimal
prime ideal or a maximal ideal and in C(Y) there exists a prime z^{°}ideal which is neither a minimal prime ideal nor a maximal ideal.
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