Winter 2001

Do all problems. All rings should be assumed to have a 1.

- Let
*R*,*A*and*B*be commutative rings with*R**A*and*R**B*. Prove that if*A*is an integral extension of*R*and*B*is an integral extension of*R*, then the ring*A**B*is also an integral extension of*R*. - For this problem all fields have characteristic 0. Let
*K*/*L*be a Galois extension with Galois group*G*and let*H*be a subgroup of*G*. Prove that there exists some b*K*such that*H*coincides with{s*G*| s(b) = b}. - Let
*S*be a semisimple ring. Prove that*S*X*S*is semisimple. - Let
*G*be a finite group and assume that*p*is a fixed prime divisor of its order. Set*K*= N_{G}(*P*) where the intersection is taken over all Sylow*p*-subgroups*P*of*G*and N_{G}() denotes the normalizer. Show that- (a)
*K**G*.- (b)
*G*and*G*/*K*have the same number of Sylow*p*-subgroups.

- Suppose
*A*is an abelian group (written additively) of order*p*^{M}for some prime*p*. Prove that if*n*is a positive integer such that*p*^{n}*A*= 0, then|{*a**A*|*pa*= 0}|*p*^{M/n}. - Let
*G*be a finite group. Prove that if*H*and*K*are normal nilpotent subgroups of*G*, then so is*HK*. - Prove or disprove: let
_{6}denote the ring of integers modulo 6. Then every projective_{6}-module is free.