Exercise 21, § 2

Let $S$ be a multiplicative monoid with identity element $e$, satisfying condition (D) of no. 10 and further satisfying the two following conditions: (1) the relation $st = e$ implies $s = t = e$: (2) for all $s \in S$, the number $n$ of terms in a finite sequence $(t_i)_{1 \leq i \leq n}$ of elements distinct from $e$ and such that $t_1 t_2 \ldots t_n = s$ is less than a finite number $\nu(s)$ depending only on $s$.

Show that, for an element $x = \sum_{s \in S} \alpha_s s$ of the total algebra of $S$ over a ring $A$ to be invertible, it is necessary and sufficient that $\alpha_e$ be invertible in $A$ (reduce it to the case where $\alpha_e = 1$ and use the identity