Exercise 13, § 5

Let $A$ be a (commutative) ring and $B$ an $A$-algebra (not necessarily commutative). Suppose that the $A$-module $B$ is faithful and generated by a finite number of elements $b_i$ ($1 \leq i \leq n$); recall that $B^0$ denotes the opposite algebra of $B$ and that a canonical homomorphism is defined of the $A$-algebra

B \otimes_A B^0 to the A-algebra End_A(B) mapping $b \otimes b'$ to the endomorphism $z \mapsto bzb'$.

(a) Show that the two following conditions are equivalent: (α) The $b_i$ form a basis of the A-module B and the canonical homomorphism $B \otimes_A B^0 \to \mathrm{End}_A(B)$ is bijective. (β) The matrix $U = (u_{ij})$ such that $u_{ij} = b_j b_i$ is invertible in $M_n(B)$.

(b) Suppose further that A is a local ring with maximal ideal $m$; let $\beta_i$ denote the canonical image of $b_i$ in $B/mB$. Show that conditions (α) and (β) are then equivalent to each of the conditions: (γ) The $\beta_i$ form a basis of the A/m-module $B/mB$ and $B/mB$ is a simple central A/m-algebra (Algebra, Chapter VIII, § 5, no. 4). (δ) The matrix $(\beta_j \beta_i)$ is invertible in $M_n(B/mB)$. (To prove that (δ) implies (β), use the fact that $mB$ is contained in the Jacobson radical of B and Exercise 5 of Algebra, Chapter VIII, § 6).