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An ideal $I$ in a commutative ring $R$ is called a $z^\circ$-ideal if $I$ consists of zero-divisors and for each $a\in I$ the intersection of all minimal prime ideals containing $a$ is contained in $I$. We
characterize topological spaces $X$ for which $z$-ideals and $z^\circ$-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 $P$-spaces are
characterized in terms of $z^\circ$-ideals. Finally, we construct two topological almost $P$-spaces $X$ and $Y$ which are note $P$-spaces and such that in $C(X)$ every prime $z^\circ$-ideal is either a minimal
prime ideal or a maximal ideal and in $C(Y)$ there exists a prime $z^\circ$-ideal which is neither a minimal prime ideal nor a maximal ideal.
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