Countable strongly annihilated ideals in commutative rings

In this paper we introduce and study the concept of countable strongly annihilated ideal in commutative rings, in particular in rings of continuous functions. We show that a maximal ideal in $C(X)$ is countable strongly annihilated if and only if it is a real maximal $z^\circ$-ideal. It turns out that $X$ is an almost $P$-space if and only if countable strongly annihilated ideals and strongly divisible $z$-ideals coincide if and only if every $M_x$ is a countable strongly annihilated ideal, for any $x\in X$. We observe that an almost $P$-space $X$ is Lindelof if and only if every countable strongly annihilated ideal is fixed. We give a negative answer to a question raised by Gilmer and McAdam.