MATH 341 Final, Fall 2004

. = ON wea, i ee \ ee ot ee \Oe . - Math 341 Final Examination N. Nahlus Time: 2 hours Jan 27, 2005 Part I (80%) Answer 1-3 AND any other 5 problems of the following 10 problems. 1, Overa p.id R, show that every submodule of R” is free of rank at most n. 2. Prove or disprove by _a counter example. a) Over a p.id R, a torsion free module is free. b) If A®, B=0, then A=0 or B=0. 3. Prove or disprove by _a counter example, a) If a®, b=0 (in A®, B),, Ais free and B is torsion free, then a=0 or b=0. b) If4@, B=0, A’ <A, then A'S, B=0. 4. Prove or disprove by _a counter example. a) Let f s A> Z,, be a surjective morphism of abelian groups, and let g: Z > Z,, bea morphism of abelian groups, then there exists a morphism 4: Z— A suchthat foh=g. b) Over a domain D, a free module is torsion free. c) Over an integral domain D, let T be the submodule of torsion elements of a module M, then M’T is torsion free 5. Let f:M — M bean epimorphism of an Noetherian module M. Show that f is an isomorphism. 6. Use Zorn’s Lemma to show that every proper submodule of a finitely generated R-module is contained in a maximal submodule. 7. Prove or disprove by _a counter example. a) Every submodule of a Nocthcrian module is also Noetherian. b) Every subring of a Noetherian ring is also Noetherian. c) Every Artinian integral domain is a field 8. (i) State very carefully 3 equivalent definitions for a projective module and (ii) prove any (non-obvious) implication of the equivalence. (iii) Is Q a projective Z-module ? Justify 9. Prove or disprove by _a counter example. a) Every submodule of a projective module is projective. b) Every quotient module of a projective module is projective. c) If every R-module is injective, then every R-module is projective. 10. Use Nakayama’s Lemma, to prove that every finitely generated projective module over a local ring is free. Part 2. (20%) See next page Part 2. (20%) 1) Prove one part of the following (a) Let M be the ring of n by n matrices over a left noetherian ring R. Let S_be the subring of M ofall the upper triangular matrices. Show that S is a left noetherian ring. (b) Show that division rings are the only rings over which every module is free. (Hint: Recall that R is a division ring iff R has no non-trivial maximal left ideals.) 2) Show that A @ B is projective <> A and Bare projective 3) Let 03 A> BC +0 bean exact sequence of modules (over a ring R). Prove or disprove (by a counter example) (a) If A and B are projective, then C is projective. (b) If A and C are projective, then B is projective. (c) If B and C are projective, then A is projective.