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For a finite group $G$, (semi-)Mackey functors and (semi-)Tambara functorsare regarded as $G$-bivariant analogs of (semi-)groups and (semi-)ringsrespectively. In fact if $G$ is trivial, they agree with the ordinary(semi-)groups and (semi-)rings, and many naive algebraic properties concerningrings and groups have been extended to these $G$-bivariant analogous notions.  In this article, we investigate a $G$-bivariant analog of the semi-grouprings with coefficients. Just as a coefficient ring $R$ and a monoid $Q$ yieldthe semi-group ring $R[Q]$, our constrcution enables us to make a Tambarafunctor $T[M]$ out of a semi-Mackey functor $M$, and a coefficient Tambarafunctor $T$. This construction is a composant of the Tambarization and theDress construction.  As expected, this construction is the one uniquely determined by therighteous adjoint property. Besides in analogy with the trivial group case, if$M$ is a Mackey functor, then $T[M]$ is equipped with a natural Hopf structure.  Moreover, as an application of the above construction, we also obtain some$G$-bivariant analogs of the polynomial rings.

A generalization of The Dress construction for a Tambara functor, and polynomial Tambara functors