Ricci almost solitons on semi-Riemannian warped products

It is shown that a gradient Ricci almost soliton on a warped product, (B-n x(h) F-m,g,f,lambda) whose potential function f depends on the fiber, is either a Ricci soliton or lambda is not constant and the warped product, the base and the fiber are Einstein manifolds, which admit conformal vector fields. Assuming completeness, a classification is provided for the gradient Ricci almost solitons on warped products, whose potential functions depend on the fiber. An important decomposition property of the potential function in terms of functions which depend either on the base or on the fiber is proven. In the case of a complete gradient Ricci soliton, the potential function depends only on the base.