**Algebra Prelim Spring 1994
Answer All Problems**

*G*be a group with exactly three elements of order two. Prove that*G*is not simple.*G*has a homomorphic image isomorphic to*S*_{3}(the symmetric group of degree 3), but is*not*isomorphic to*S*_{3}.*R*be a PID which is*not*a field, and let*M*be a finitely generated*R*-module which is*not*a torsion module.(i) Prove that the

*R*-module*R*is isomorphic to a proper submodule of itself.(ii) Prove that

*M*is isomorphic to a proper submodule of itself.*R*be a ring, and let*A*,*B*,*C*be*R*-modules.(i) Prove that as abelian groups.

(ii) Prove that is

*not*isomorphic to .*R*be a commutative Noetherian ring.(i) If

*S*is a multiplicative subset of*R*, prove that*S*^{-1}*R*is Noetherian.(ii) Prove that

*R*[[*X*^{-1},*X*]] (the Laurent Series ring in*X*) is a Noetherian ring (you may assume that the power series ring*R*[[*X*]] is Noetherian).*X*^{4}+*X*^{3}+*X*^{2}+*X*+1 is irreducible in . (Set*Y*=*X*-1.)(ii) Let , and suppose no eigenvalue of

*A*is equal to 1. Prove that if*A*^{5}=*I*, then 4|*n*(where*I*denotes the identity matrix).

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Tue Jul 30 12:37:23 EDT 1996