Exercise 21, § 5

Let A be a ring and n an integer $\geqslant 0$.

(a) Show that A-modules F such that $F^n$ is isomorphic to $A^n$ are projective of rank 1 and that their classes in $P(A)$ form a subgroup $P_n(A)$ every element of which is annihilated by $n$. (To show the first assertion, use Proposition 5 of § 2, no. 2; for the second, observe that $\wedge^n F^n$ and $\bigotimes^n F$ are canonically isomorphic.)

(b) Let $\mathrm{Aut}_n(A)$ be the group of automorphisms of the $A$-algebra of matrices $\mathbf{M}_n(A)$ and let $\mathrm{Int}_n(A)$ be the subgroup of inner automorphisms of $\mathbf{M}_n(A)$. Show that the quotient group $\mathrm{Aut}_n(A)/\mathrm{Int}_n(A)$ is isomorphic to $P_n(A)$ (use Exercise 20 with $P = P' = A^n$ or Exercise 9 of Algebra, Chapter VIII, ). Deduce that, if $A$ is a semi-local ring or a principal ideal domain, then

\* (c) If $A$ is a Dedekind domain, show that $P_n(A)$ is identical with the subgroup of $P(A)$ consisting of the elements annihilated by $n$.