Exercise 6, § 6

Let A be a unital $\mathbf{Z}$-algebra. We suppose that A is a free $\mathbf{Z}$-module. Let $\mathscr{B}$ be a subset of A which is a basis of it as a $\mathbf{Z}$-module. We say that the pair $(A,\mathscr{B})$ is a natural algebra if the structure constants of A with respect to $\mathscr{B}$ (A, III, p. 10) belong to $\mathbf{N}$ and if the coefficients of $1_A$ in the basis $\mathscr{B}$ belong to $\mathbf{N}$. We say moreover that $(A,\mathscr{B})$ is united if $1_A$ is an element of $\mathscr{B}$.

Let $(A,\mathscr{B})$ be a natural algebra. Let $\mathscr{B}_0$ be the set of the $b\in \mathscr{B}$ such that the coefficient of $b$ in $1_A$ is not zero. We denote by $\tau_A$ the unique morphism of $\mathbf{Z}$-modules from A to $\mathbf{Z}$ such that, for every $b$ in $\mathscr{B}$, one has $\tau_A(b) = 1$ if $b\in \mathscr{B}_0$, and $\tau_A(b) = 0$ otherwise.

a) One has $\tau_A(xy) =\tau_A(yx)$ for all $(x, y)\in A^2$.

b) For all $b\not =b'$ in $\mathscr{B}_0$, one has $b^2=b$ and $bb'= 0$.