A module M is called a \(C_{41}\) -module if whenever A is a nonsingular submodule of M and B is a direct summand of M with A isomorphic to B and \(A\cap B = 0\) , then A is a direct summand of M. This concept is a generalization of the notion of \(C_4\) -modules. We show that direct sums of two \(C_{41}\) -modules do not inherit the property, in general. The class of rings R for which any arbitrary direct sum of \(C_{41}\) -modules is also a \(C_{41}\) -module is shown to be exactly that of right t-semisimple rings. We proved that the class of rings R for which every \(C_{41}\) -module satisfies the \(C_4\) -condition is precisely that of right SI-rings. We also investigate the rings R, whose all finitely generated free R-modules satisfy the \(C_{41}\) -condition.