Algebra Prelim Spring 1994
Answer All Problems
(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.
(i) Prove that as abelian groups.
(ii) Prove that is not isomorphic to .
(i) If S is a multiplicative subset of R, prove that S-1R 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).
(ii) Let , and suppose no eigenvalue of A is equal to 1. Prove that if A5 = I, then 4|n (where I denotes the identity matrix).