Consider a morphism of commutative monoids $u\colon M\rightarrow N$. We say that $u$ is *flat,* if the tensor product functor $\bullet\otimes\_MN$ from the category of $M$-modules to the category of $N$-modules commutes with finite projective limits.


Let $R$ be a non-zero commutative ring, and suppose that the induced morphism of $R$-algebras $R[u]\colon R[M]\rightarrow R[N]$ is flat.



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Descent of flatness from algebras to monoids
Here is an easier answer. Let $R$ be any non-zero commutative ring with unit. Let $G$ be any group and let $H$ be a proper subgroup. Let $u\colon H\to G$ be the inclusion. Then $u$ is never flat but $RG$ is a free $RH$-module and hence flat. The problem is that the terminal $H$-set $1$ tensored with $G$ over $H$ yields $G/H$ as a $G$-set, which is not there terminal $G$-set. 


