Compact almost Ricci solitons with constant scalar curvature are gradient

The aim of this note is to prove that any compact non-trivial almost Ricci soliton $big (M^n,,g,,X,,lambda big )$ with constant scalar curvature is isometric to a Euclidean sphere $mathbb {S}^{n}$ . As a consequence we obtain that every compact non-trivial almost Ricci soliton with constant scalar curvature is gradient. Moreover, the vector field $X$ decomposes as the sum of a Killing vector field $Y$ and the gradient of a suitable function.