Exercise 2, § 18

Let A be a K-algebra and $m$ an integer. Suppose that every element of A is algebraic of degree $\leqslant m$ over K.

a) Prove that if A is a primitive ring, then there exist an integer $r\leqslant m$ and a field D whose elements all have degree $\leqslant m$ over K such that A is isomorphic to the matrix algebra $\mathbf{M}_r(D)$. (If A is not simple, then deduce from the density theorem that for every integer $n$, there exist a subalgebra $A_n$ of A and a surjective homomorphism $A_n\rightarrow \mathbf{M}_n(D)$. Observe that the algebra $\mathbf{M}_n(D)$ contains elements X such that $X^{n-1}\not= 0$ and $X^n= 0.$)

b) Suppose that the field K is infinite. Prove that the algebra $A/\mathfrak{R}(A)$ is semisimple and admits at most $m$ simple components (observe that the algebra $K^n$ contains elements of degree $n$ over K; deduce that A admits at most $m$ maximal two-sided ideals, and conclude using a)). Prove that if K is, moreover, perfect, then $A/\mathfrak{R}(A)$ has finite rank over K.