Uno±cial Math 501 Problems The following document consists of problems collected from previous classes and comprehensive exams given at the University of Illinois at Urbana-Champaign. In the exercises below, all rings contain identity and all modules are unitary unless otherwise stated. Please report typos, sug- gestions, gripes, complaints, and criticisms to: [email protected] 1) Let R be a commutative ring with identity and let I and J be ideals of R. If the R-modules R=I and R=J are isomorphic, prove I = J. Note, we really do mean equals! 2) Let R and S be rings where we assume that R has an identity element. If M is any (R; S)-bimodule, prove that S HomR(RR; M) ' M: 3) Let 0 ! A ! B ! C ! 0 be an exact sequence of R-modules where R is a ring with identity. If A and C are projective, show that B is projective. 4) Let R be a ring with identity. (a) Prove that every ¯nitely generated right R-module is noetherian if and only if R is a right noetherian ring. (b) Give an example of a ¯nitely generated module which is not noetherian. ® ¯ 5) Let 0 ! A ! B ! C ! 0 be a short exact sequence of R-modules where R is any ring. Assume there exists an R-module homomorphism ± : B ! A such that ±® = idA. Prove that there exists an R-module homomorphism : C ! B such that ¯ = idC . 6) If R is any ring and RM is an R-module, prove that the functor ¡ ­R M is right exact. 7) Let S and R be rings, with S a subring of R containing 1R. (a) Use tensor products to construct a covariant functor F : ModS ! ModR : 1 (b) If MS is a at module, prove that MS is S-isomorphic with a submodule of F (MS). 8) Let R be any ring and let MR, NR be at R-modules. Prove that M © N is also at. 9) Let R be a domain and let MR be a at R-module. Prove that M is R- torsion-free. 10) Give an example of a commutative ring R with identity and an R-module M such that the R-torsion elements of M do not form a submodule of M. 0 0 11) Let (C; ¹¸)¸2¤ and (C ; ¹¸)¸2¤ be coproducts of set of objects fA¸ : ¸ 2 ¤g in some category C. Prove that there is an equivalence ¯ 2 C(C; C 0) such that 0 ¹¸ = ¯¹¸ for all ¸ in ¤. 12) Prove that the direct sum ¸2¤ M¸; ¹¸ is the coproduct in the category R Mod where ¹¸ : M¸ ! M¸ via ¸2¤¡L ¢ L a if º = ¸ (¹¸(a))º = (0Mº if º 6= ¸: 13) Prove that HomR(¡; M) is a contravariant functor from R Mod to Ab. 14) Let R and S be rings and ® : R ! S a ring homomorphism. Show that there is a corresponding covariant functor F® : S Mod ! R Mod. 15) Let L; M; N be submodules of an R-module and assume that N M. Prove that L + M=L + N is R-isomorphic with a quotient of M=N and L \ M=L \ N is R-isomorphic with a submodule of M=N. 0 0 R 0 16) Let (F; ¹) and (F ; ¹ ) be free on a set X in R Mod. Prove that F ' F . 17) In the following diagram the squares are commutative, the rows are exact, and ¸ and º are isomorphisms of modules. All maps are module homomorphisms. ® ¯ A B C 0 ¸ ¹ º 0 ®0 ¯ 0 A0 B0 C0 Prove: (a) ® is injective. (b) ¯0 is surjective. 2 (c) ¹ is an isomorphism. 18) Let R be a ring with identity. Prove that every R-module is projective if and only if R is a semisimple ring. 19) Let R be an arbitrary ring and let M be an R-module with submodules M1; M2; : : : ; Mk such that each M=Mi is a simple module and M1 \ M2 \ ¢ ¢ ¢ \ Mk = f0g: Prove that M is a semisimple module. 20) Let G be a group and F a ¯eld. De¯ne a map : F G ! F by fgg = fg: 0 1 g2G g2G X X @ A (a) Prove that is a ring homomorphism. (b) Prove that Ker() = IG where IG is generated as an F -vector space by the set fg ¡ 1 : g 2 Gg. (c) Assume that G has ¯nite order not divisible by char(F ). Prove F G = J0 © J1 where J0; J1 are ideals of F G and IGJ0 = 0, IGJ1 = J1. (d) Show that dimF (J0) = 1 and ¯nd a generator for J0. 21) Let G be a nontrivial ¯nite p-group and let F be a ¯eld of characteristic p, a prime. Let IG be as in the above problem. Prove IG is not a direct summand of the module F G(F G). What does this tell you about the ring F G? 22) Let R be a PID and denote by F the ¯eld of fractions of R. Regard F as an R-module in the obvious way. (a) If X is a ¯nitely generated R-submodule of F , show that X = Rf for some f 2 F . (b) If F is a ¯nitely generated R-module, prove R = F . 23) Let M be a ¯nitely generated module over a PID R. (a) If M is a torsion module prove that M is artinian. (b) Assuming that R is not a ¯eld, show that M has a composition series if and only if it is a torsion module. 3 24) Describe all isomorphism types of abelian groups of order 2250, and give the invariant factors for each type. 25) Let M and N be ¯nite abelian p-groups and let their invariant factors be p; : : : ; p; p2; : : : ; p2; : : : ; pr; : : : ; pr; m1 m2 mr and | {z } | {z } | {z } p; : : : ; p; p2; : : : ; p2; : : : ; pr; : : : ; pr; n1 n2 nr respectively, where mi;|ni{z> }0.| What{z }are the| in{zvarian} t factors of M ­Z N? Justify your answer. 26) Prove that an artinian integral domain is ¯eld. [Use the DCC directly, no major theorems necessary.] 27) Let R be a ring and M a left R-module. If r 2 R and a 2 M, de¯ne a ¢ r to be r ¢ a. (a) Prove that the assignment (a; r) 7! a ¢ r makes M into a right module over ROpp. (b) De¯ne a covariant functor Opp : R Mod ! ModROpp . 28) Give an example of a nonzero abelian group A such that A ­Z A = 0. 29) Let M¸ be a right R-module for ¸ 2 ¤ and let N be a left R-module, where R is any ring. Prove that M¸ ­R N ' (M¸ ­R N) : à 2¤ ! 2¤ M¸ M¸ ­n 30) Let A = Z©Z2©Z3©Z3. Let A denote the nth tensor power, A­Z¢ ¢ ¢­ZA, (n factors). Prove that ­n 2n¡1 3n¡1 A ' Z © Z2 © Z3 ; (n > 0); m where Zk = Zk © ¢ ¢ ¢ © Zk; (m factors). 31) Let R be a commutative ring and let M be an R-module. The exterior square M ^R M is an abelian group generated by elements a ^ b, (a; b 2 M), with all the properties of the tensor product plus a ^ a = 0; for all a 2 M. (a) Explain how to de¯ne M ^R M. 4 (b) Prove that a ^ b = ¡(b ^ a), where a; b 2 M. (c) A mapping ® : M £ M ! A, where A is an abelian group, is called alternating middle linear if it is middle linear and also ®((a; a)) = 0; for all a 2 M: Construct a \canonical" alternating middle linear mapping : M £ M ! M ^R M: (d) Use alternating middle linear maps to characterize M ^R M up to isomor- phism by a mapping property. 32) Let R be an arbitrary ring with identity. (a) Prove that a left R-module M is simple if and only if M ' R=I for some maximal left ideal I. (b) Use Zorn's Lemma to prove that simple left R-modules exist. 33) Let 0 0 0 ¡1 1 0 0 4 A = 2 3 0 1 0 ¡6 60 0 1 4 7 6 7 be a matrix with entries in a ¯eld 4k of characteristic5 p > 5. (a) Prove that A is similar to a Jordan block J. (b) Find the order of A in the group GL4(k). 34) Let R be a commutative ring and let R Mod be the category of all R- modules. (a) If M is an R-module, prove that R ­R M is also an R-module. (b) Prove that M ' R ­R M. (c) Prove that I , the identity functor on R Mod, is naturally equivalent to the functor R ­R ¡. 35) De¯ne semisimple ring, and prove that a commutative ring R is semisimple if and only if R is isomorphic to a ¯nite direct product of ¯elds. 36) Let S2 = f(a; b; c) 2 R3 : a2 + b2 + c2 = 1g be the unit sphere in R3. Prove that I = ff(x; y; z) 2 R[x; y; z] : f(a; b; c) = 0 for all (a; b; c) 2 S2g is a ¯nitely generated ideal in R[x; y; z]. 5 37) Prove that a UFD satis¯es the ascending chain condition for principal ideals, but that the ascending chain condition on all ideals need not hold. 38) Let R be a ¯nitely generated commutative ring with identity. Prove that R is isomorphic with some quotient ring of a polynomial ring Z[t1; : : : ; tn].