In this work g-radical supplemented modules are defined which generalize gsupplemented modules. Some properties of g-radical supplemented modules are investigated. It is proved that the finite sum of g-radical supplemented modules is g-radical supplemented. It is also proved that every factor module and every homomorphic image of a g-radical supplemented module is g-radical supplemented. Let R be a ring. Then R R is g-radical supplemented if and only if every finitely generated R-module is g-radical supplemented. In the end of this work, it is given two examples for g-radical supplemented modules separating with g-supplemented modules.

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In this study, we consider some power series with rational coefficients and investigate transcendence of the values of these series for Liouville number arguments. It is proved that these values are either a Liouville number or a rational number under certain conditions.

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In this work, some new properties of (amply) g-radical supplemented modules are investigated. It is proved that every factor module and every homomorphic image of an amply g-radical supplemented module are amply g-radical supplemented. Let M be a π-projective and g-radical supplemented module. Then M is amply g-radical supplemented. Let M be a projective and g-radical supplemented module. Then every finitely M-generated module is amply g-radical supplemented. Let R be any ring. Then R R is g-radical supplemented if and only if every finitely generated R-module is amply g-radical supplemented.