K. Morita proved that the extension of scalars functor for a morphism of rings $f:R\to S$ is Frobenius iff the morphism $f$ itself is Frobenius in the sense of (Kasch), that is: ${}_R S$ is finitely generated projective and ${}_S S_R\cong Hom_R({}_R S, {}_R R)$ as $R-S$-bimodules.

This is in the spirit of the finite-dimensional duality coded e.g. in the notion of Frobenius algebra.

References

Stefaan Caenepeel, Gigel Militaru, Shenglin Zhu, Frobenius and separable functors for generalized module categories and nonlinear equations, Springer Lec. Notes in Math. 1787 (2002) xiv+354 pp, gBooks