  We investigate the triviality of compact Ricci solitons under general scalar
conditions involving the Weyl tensor. More precisely, we show that a compact
Ricci soliton is Einstein if a generic linear combination of divergences of the
Weyl tensor contracted with suitable covariant derivatives of the potential
function vanishes. In particular we recover and improve all known related
results. This paper can be thought as a first, preliminary step in a general
program which aims at showing that Ricci solitons can be classified finding a
"generic" $[k, s]$-vanishing condition on the Weyl tensor, for every $k,
s\in\mathbb{N}$, where $k$ is the order of the covariant derivatives of Weyl
and $s$ is the type of the (covariant) tensor involved.
