The structure of algebras admitting well agreeing near weights

We characterize algebras admitting two well agreeing near weights $\rho$ and $\sigma$. We show that such an algebra $R$ is an integral domain whose quotient field $\mathbf{K}$ is an algebraic function field of one variable. It contains two places $p,Q\in \mathbb{P}(\mathbf{K})$ such that $\rho$ and $\sigma$ are derived from the valuations associated to $P$ and $Q$. Furthermore $\bar R= \cap_{S\in{\mathbb P}(\mathbf F)\setminus{P,Q}}{\mathcal O}_S$.