Exercise 9, § 13

Let K be a commutative field and A a K-algebra. Suppose that for every Artinian K-algebra B, the ring $A\otimes_KB$ is Artinian. Prove that A is finite-dimensional over K (reduce to the case when A is semisimple by observing that the $A/\Re (A)$-modules $\Re (A)^k/\Re (A)^{k+1}$ have finite length; then use Exercises 6 and 7).