Exercise 5, § 8

Let M be a monoid, $\mathbf{Z}^{(M)}$ the free commutative group with basis M and $u : M \to \mathbf{Z}^{(M)}$ the canonical mapping.

(a) Show that there exists a unique multiplication on $\mathbf{Z}^{(M)}$ such that $\mathbf{Z}^{(M)}$ is a ring and such that $u$ is a monoid morphism of the monoid M into the multiplicative monoid of the ring $\mathbf{Z}^{(M)}$.

(b) Let A be a ring and $v : M \to A$ a monoid morphism of the monoid M into the multiplicative monoid of A. There exists a unique ring morphism $f : \mathbf{Z}^{(M)} \to A$ such that