Algebra II. Ring Theory: Ring Theory by Carl Faith

By Carl Faith

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Then the following properties hold: (1) every left (resp. right) A-module M has a natural structure of a k-vector space, induced by scalar multiplication. Moreover, M is finite dimensional over k whenever it is finitely generated as an A-module; (2) A has a minimal left (resp. right) ideal; (3) for all n, m ≥ 1, we have An ∼ =A Am ⇐⇒ n = m. Proof. The first point is obvious. To prove (2), observe that any left (resp. right) ideal of A is in particular a linear subspace of A, hence finite dimensional over k since A is.

Proof. 8, so D contains R. Notice also that Z(D) is a field containing R, so Z(D) is isomorphic to either R or C, since Z(D)/R has finite degree by assumption. If D is commutative, then D is a field extension of R, hence is isomorphic to R or C. Assume now that D is not commutative, and let k = Z(D). Then k cannot be isomorphic to C. Otherwise, for all d ∈ D, the minimal polynomial μd,k of d would be of degree 1, meaning that d ∈ k. We then would have that D = k, contradicting the fact that D is not commutative.

Since L is a simple k-subalgebra of A and ZA (L) = L (since L is commutative), the last part of the Centralizer Theorem gives ∼L Mr (L ⊗L CA (L)). A ⊗k L = Since L ⊗L CA (L) ∼ =L CA (L), we are done. Exercises 1. Let A be a central simple k-algebra. Show that the group of all k-algebra automorphisms of A is canonically isomorphic to A× /k× . 2. Let A be a simple k-algebra, not necessarily central. Is every automorphism of A inner? 3. The goal of this exercise is to provide an elementary and totally explicit proof of Skolem-Noether’s theorem for matrix algebras.

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