The minimal prime decomposition for semiprime ideals is defined and studied on
z-ideals of C(X). The necessary and sufficient condition for existence of the minimal prime decomposition of a z-ideal / is given, when / satisfies one of the following conditions: (i) / is an intersection of maximal ideals. (ii) I is an intersection of O , s, when X is basically disconnected. (iii) I=O , when x X has a countable base of neighborhoods. (iv) I is finitely generated. (v) / is countably generated, when X is
compact and countable of first kind