It is shown that if R is a ring such that every countably generated right R-module is a direct sum of a projective module and an injective module then R is right Artinian and every singular right (or left) R-module is injective. The proof is as follows. The ring R is proved right Noetherian by showing that every cyclic right R-module is a direct sum of a projective module and a Noetherian module, and by then applying a theorem of A. W. Chatters [Q. J. Math., Oxf. II. Ser. 33, 65-69 (1982; Zbl 0443.16011)]. Next R is shown to be right Artinian by showing that the injective hull of the right R-module R is finitely generated [see C. I. Vinsonhaler, J. Algebra 17, 149-151 (1971; Zbl 0211.362)]. The rest follows easily.