\textit{S. Singh} [Arch. Math. 39, 306-311 (1982; Zbl 0502.16012)] considered rings \(R\) with the property: (P) every finitely generated right \(R\)-module is a direct sum of a projective module with zero socle and uniserial Artinian modules. He proved that a right FBN-ring satisfying (P) is a direct sum of an Artinian serial ring and right hereditary prime rings. In this note the authors prove that a ring \(R\) is Noetherian serial if and only if \(R\) is semiperfect and satisfies the following condition: (P') every finitely generated right \(R\)-module is a direct sum of a projective module with zero socle and uniserial modules with finite length. One consequence is that any right nonsingular ring satisfying (P) is a direct sum of an Artinian serial ring, a serial ring with zero socle and prime rings. Finally, it is shown that a ring \(R\) is Artinian serial if and only if \(R\) satisfies (P) and \(eR\) is not isomorphic to a proper submodule of itself, for any primitive idempotent e, and this is equivalent to \(R\) having the following two properties: (a) every cyclic right \(R\)-module is a direct sum of an injective module and uniserial modules with non-zero socles, and (b) for any minimal right ideal \(S\) of \(R\), every 2-generated submodule of its injective hull \(E(S)\) is either projective or singular.