Monoid ring

Remarks

Example: for the monoid $M = \set{1, x, x^2, \dots}$ with $x^i x^j = x^{i+j}$, the convolution is the usual product of polynomials, so $R[M]$ is the polynomial ring $R[x]$. For instance $(1 + x)(1 - x) = 1 \cdot 1 + (1 - 1) x - x^2 = 1 - x^2$. For a group $G$, $R[G]$ is the group ring, and for $M = (\Z, +)$ it is the Laurent polynomial ring. Non-example: the set of all functions $\Z \to R$ with the same product is not a ring, because the finiteness condition fails and the convolution sum $\sum_{i + j = n} f(i) g(j)$ can be an infinite sum. The ring $R[M]$ is commutative when $R$ and $M$ are commutative. When $R$ is commutative, each monoid homomorphism from $M$ to the multiplicative monoid of an $R$-algebra $A$ extends uniquely to an $R$-algebra homomorphism $R[M] \to A$. In additive notation for $M$, the same ring is often called the monoid algebra.