Exercise 11, Appendix 1

Suppose that $k$ is a field, that A is a $k$-algebra, and that every decreasing sequence of subalgebras of A is stationary.

a) Prove that the algebra A is Artinian and that its radical is finite-dimensional over $k$ (apply Exercise 9).

b) Let S be a simple component of $A/\Re (A)$ (Exercise 7); it is isomorphic to a matrix algebra over a field D, with center Z. Prove that D is finite-dimensional over Z (if D contains an element that is transcendental over Z, construct an infinite strictly decreasing sequence of subfields of D; if D is algebraic and infinite-dimensional over Z, construct an infinite strictly increasing sequence of subfields of D, and consider their commutants in D).

c) Prove that if Z is infinite-dimensional over $k$, then we have S = D (consider the nilpotent subalgebras of B).

d) Prove that Z is an algebraic extension of $k$. Let $p$ be the characteristic exponent of $k$ and L the largest separable extension of $k$ contained in Z. Prove that $L\cap Z^p$ has finite degree over $L^p$ (consider a $p$-basis of $L\cap Z^p$ over $L^p$).

e) Every decreasing sequence of subextensions of L is stationary. Give an example to show that this condition does not imply that L has finite degree over $k$ (cf. V, §12, p. 167, Exercise 3). f ) Conversely, let B be a $k$-algebra satisfying the conditions of a) such that every simple component of $B/\Re (B)$ satisfies the conditions of b), c), d), and e). Prove that every decreasing sequence of subalgebras of B is stationary.

(Reduce to the case when B is a field, with center Z. Let $(D_n)$ be a decreasing sequence of subfields of B; we may assume that the field D generated by $D_n$ and Z is independent of $n$. Observe that we can write $D = E\otimes_LZ$, where L is a subfield of Z of finite degree over $k$ and E is a field of finite degree over L whose center contains L. Prove that $D_n$ is then the field generated by E and $D_n\cap Z$. We are thus reduced to the case when B is a commutative field that is a $p$-radical extension of $k$. If $(F_n)$ is a decreasing sequence of subextensions of B, consider the subfield $k(F^{p^{\infty}}_n)$, where $F^{p^{\infty}}_n=\cap_fF^{p^f}_n.$)