Monoid ring

Summary

The monoid ring $R[M]$ of a monoid $M$ over a ring $R$ is the ring of finite formal sums $\sum r_m m$, multiplied with the monoid law. It contains $R$ and a copy of $M$ inside its multiplicative monoid. Polynomial rings, Laurent polynomial rings and group rings are all monoid rings. It is the free way to turn a monoid into a ring with coefficients in $R$.