Exercise 5, § 12

Let A be a quasi-simple K-algebra (VIII, p. 120, Remark 2).

a) Prove that the $(A$, A)-bimodule $_sA_d$ is simple; deduce that the center Z of A is a field, isomorphic to the commutant of this bimodule.

b) Let B be a K-algebra. Prove that every two-sided ideal of $A\otimes B$ is of the form $A\otimes_Z\mathfrak{b}$, where $\mathfrak{b}$ is a two-sided ideal of B (reduce to the case Z = K, and use Corollary 2 of VIII, p. 63).

c) Deduce that if A and B are quasi-simple K-algebras of which one has center K, then the algebra $A\otimes B$ is quasi-simple.

d) Let B be a K-algebra and A a quasi-simple subalgebra of B whose center Z contains K. Prove that A and its commutant $A'$ in B are linearly disjoint over Z (use b)).

e) Let A be a quasi-simple K-algebra and Z its center. Let $\varphi : A\otimes A^o\rightarrow$ End$_Z(A)$ be the homomorphism that sends $a\otimes b$ to the endomorphism $x\mapsto axb$. Prove that $\varphi$ is injective and that its image H is a quasi-simple algebra (use c) and d)). Prove, moreover, that H is dense in End$_Z(A)$ (VIII, p. 129, Exercise 3); we have H = End$_Z(A)$ if and only if A is finite-dimensional over Z (cf. VIII, p. 128, Exercise 2). f ) Let A be a semisimple algebra with center K. Deduce from e) that the K-algebra $A\otimes A^o$ is semisimple if and only if A has finite rank over K (cf. VIII, p. 238, Theorem 2).