MATH 535 HOMEWORK ASSIGNMENT 1

due Feb. 6

1. For sequence of modules over a unital commutative ring R, M M M 0, prove it is exact iff for any R-module N, the induced sequence 0 HomR M N HomR M N HomR M N is exact. 2. Let R be a domain (unital commutative ring without zero-divisor), M be a R-module, let T M m M r R rm 0. a). Prove that T M is a submodule of M. b). Prove that for any exact sequence 0 A B C, 0 T A T B T C is exact too. c). Prove or disprove If B C 0 is exact, so is T B T C 0. 3. (Universal property of direct sum and direct product) Let Mi be a family of modules over commutative ring R, N be a R-module. a). If fi : Mi N is a family of R-homomorphism, then there exists a unique R-homomorphism f : Mi N such that for any i, f i fi , where i is the standard embedding Mi Mi. b). If gi : N Mi is a family of R-homomorphism, then there exists a unique R-homomorphism f : N Mi such that for any i, pi g gi , where pi is the standard projection Mi Mi. 4. Let Mi i I, N be modules over a unital commutative ring R, prove that a). Hom Mi N Hom Mi N , b). Hom N Mi Hom N Mi. 5. Let Pi i I be modules over a unital commutative ring, then Pi is projective iff each Pi is projective. 6. Prove the equivalence (R is unital and commutative): a). 0 A B C 0 is splitting short exact sequence of R-modules.
b). For any R-module D, 0 Hom D A Hom D B Hom D C 0 is splitting short exact sequence. c). For any R-module D, 0 Hom C D Hom B D Hom A D 0 is splitting short exact sequence. 7. Prove that for a nitely generated projective R-module, Hom P R is a nitely generated projective R-module. 8. If we denote Hom M R by M , prove that for a nitely generated projective R-module P, P P. 9. Prove or disprove that if F is a free R-module, M is a submodule of F, then for any R-module N and R-homomorphism f : M N, it can be extend to a homomorphism F N 10. Prove that any nitely generated projective -module is free. 11. Prove A R B R C A R B R C for modules A, B, C over a unital commutative ring R. 12. Prove that m mn. n Hom V V. 13. For vector space V over eld k, prove that V k V 14. Prove the equivalence (R is unital and commutative): a). 0 A B C 0 is splitting short exact sequence of R-modules. b). For any R-module D, 0 D A D B D C 0 is splitting short exact sequence.

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