A fresh look into monoid rings and formal power series rings

TL;DR

This paper explores monoid rings and formal power series rings, offering new universal properties and systematic approaches to understanding polynomial and power series structures.

Contribution

It introduces a unified framework using monoid rings to derive universal properties and proofs for polynomials and formal power series.

Findings

Universal property for formal power series rings established

Systematic approach simplifies proofs of polynomial and power series facts

Provides canonical methods for studying monoid and polynomial rings

Abstract

In this article, the ring of polynomials is studied in a systematic way through the theory of monoid rings. As a consequence, this study provides natural and canonical approaches in order to find easy and rigorous proofs and methods for many facts on polynomials and formal power series; some of them as sample are treated in this note. Besides the universal properties of the monoid rings and polynomial rings, a universal property for the formal power series rings is also established.