Let R be a ring, and let X be a class of right R-modules which contains the zero module, is closed under isomorphism, and is such that every module in X has finite uniform dimension. The author investigates the situation in which every cyclic right R-module is the direct sum of a projective module and a module in the class X. A characterisation of such rings with no non-zero right ideals in the class X is given in terms of triangular matrices. This is then specialised to the case in which R is right Noetherian and X contains all simple right R-modules. In particular it is shown that every cyclic right R-module is the direct sum of a projective module and a module of finite length if and only if R is right Noetherian and has a two-sided ideal I with the following properties: \(I=eR\) for some idempotent e; I is Artinian as a right R-module; and the ring R/I is a direct sum of prime right CS rings with right Krull dimension 1.