Let $(M,g)=(N,\ddot{g})\times f(B,\bar{g})$ be an Einstein warped-product manifold (i.e. $Ric=\lambda g$) where $f:N \rightarrow (0, \infty)$ (positive scalar function) and with $g= \ddot{g}+f^2 \bar{g}$.

If $(B, \bar{g})$ is Ricci flat, being $(M, g)$ an Einstein manifold, this means that $(M, g)$ must be only Ricci flat or not? Or $(M, g)$ can be Einstein (not Ricci flat) even though $(B, \bar{g})$ is Ricci flat?