On the extension of pseudonorm and metrics to the semigroup ring

In connection with the study of algebraic systems with additional structures (topology, order, norm, etc.), it sometimes becomes necessary to continue these additional structures from the ring to their over-rings. In particular, the question arises of the possibility to extend the given additional structures of a group and a ring to their group ring.
This paper is devoted to the study of the possibility of extending a real-valued pseudonorm and a metric to the semigroup ring of a free monoid.
The main result of the paper is a theorem that establishes sufficient conditions for the possibility of extending the pseudonorm of the ring R and the metric of the space X to such a group norm on the semigroup ring RF of the ring R and the free monoid F generated by the set X such that the topology given by this group norm is annular.