Exercise 21, § 5

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

a) Now prove that the 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 $P_n(A)$ whose every element is annihilated by n. (To prove the first assertion, use Proposition 5 of § 2, No. 2; for the second, observe that $\bigwedge^n F^n$ and $\bigotimes^n F$ are canonically isomorphic.)

b) Let $\mathrm{Aut}_n(A)$ be the automorphism group of the A-algebra of matrices $\mathbf{M}_n(A)$ and let $\mathrm{Int}_n(A)$ be the subgroup of the inner automorphisms of $\mathbf{M}_n(A)$. Now prove 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 Alg., Chapter VIII, § 1). Deduce that, if A is a semi-local ring or a principal ideal domain, one has Aut_n(A) = Int_n(A).

c) When A is a Dedekind ring, now prove that $P_n(A)$ is identical with the subgroup of $P(A)$ formed by the elements annihilated by n.