Abstract
A ring R is called q-hypercyclic (hypercyclic) if each cyclic ring R-module has a cyclic quasi-injective (injective) hull. A ring R is called a qc-ring if each cyclic right R-module is quasi-injective. Hypercyclic rings have been studied by Caldwell [4], and by Osofsky [12]. A characterization of qc-rings has been given by Koehler [10]. The object of this paper is to study q-hypercyclic rings. For a commutative ring R, R can be shown to be q-hypercyclic (= qc-ring) if R is hypercyclic. (Theorems 4.2 and 4.3). Whether a hypercyclic ring (not necessarily commutative) is q-hypercyclic is considered in Theorem 3.11 by showing that a local hypercyclic ring R is q-hypercyclic if and only if the Jacobson radical of R is nil. However, we do not know if there exists a local hypercyclic ring with nonnil radical [12]. Example 3.10 shows that a q-hypercyclic ring need not be hypercyclic.