Abstract. We have introduced and investigated the notion of I-extremal disconnectedness on ideal topological spaces. First, we found that the notions of extremal disconnectedness and I-extremal disconnectedness are independent of each other. About the letter one, we observed that every open subset of an I-extremally disconnected space is also an I-extremally disconnected space. And also, in extremally disconnectedness spaces we have shown that I-open set is equivalent almost I-open and every -I-open set is preopen. Finally, we have shown that B-I-continuity( resp. pre-I-continuity, I-continuity ) is equivalent to semi-I-continuity( resp. strongly -I-continuity, almost strongly I-continuity ) if the domain is I-exteremally disconnected