Exercise 3, § 18

Let K be a commutative field. We say that a K-algebra A is algebraic if all its elements are algebraic over K. Every algebra of finite rank is algebraic; every subalgebra and quotient algebra of an algebraic algebra is algebraic. The radical of an algebraic K-algebra is a nil ideal (VIII, p. 166, Exercise 5).

a) Prove that an algebraic algebra is a pseudoregular ring (VIII, p. 178, Exercise 4; observe that for every $x\in A$, there exist an element $y\in K[x]$ and an integer $k$ such that $x^k=x^{k+1}y$).

b) Let A be an algebraic K-algebra without any nilpotent elements other than 0. Prove that A is isomorphic to a subalgebra of a product of fields. (Observe that every quotient algebra of A has the same properties. Prove that if A is primitive, then it is a field, by reasoning as in Exercise 2, a). In the general case, use Exercise 12 of VIII, p. 168.)