This short note concerns firstly with two inequalities in the geometry of gradient Ricci solitons (, f, λ) on a smooth manifold M. These inequalities provide some relationships between the curvature of the Riemannian metric and the behavior of the scalar field f through two second order equations satisfied by the scalar λ. Secondly, we propose several generalizations of Ricci solitons to the setting of manifolds endowed with linear connections, not necessary of metric type. Thirdly, we express the usual Ricci solitons equation in terms of two Golab connections