In this paper we define Z-coinitial rings, where Z is an integral domain, and prove some of their properties. In particular, we characterize commutative noetherian domains and discrete valuation domains which are Z-coinital. We define radical modules and radical rings, and we prove that every countable Z-coinitial and right hereditary ring is a right radical ring. We give some examples of rings satisfying these conditions. Finally, we prove that the lattice of preradicals of every right radical ring is not a set.