Exercise 7, § 1

Let $E$ be a monoid and $S$ a stable cancellable subset of $E$, satisfying the hypotheses of exercise 17 of III, p. 121, such that there exists a “monoid of fractions” $\overline E$ and an injective homomorphism $\varepsilon:E\to\overline E$. Let $A$ be a commutative ring. One deduces from $\varepsilon$ a homomorphism of monoid algebras $u:A^{(E)}\to A^{(\overline E)}$, which makes it possible to consider $A^{(\overline E)}$ as a left $A^{(E)}$-module. Now prove that this module is flat over $A^{(E)}$ (write $A^{(\overline E)}$ as a filtered inductive limit of free $A^{(E)}$-modules of rank 1).