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### Algebra2009Aug

Course: MATH 601, Spring 2011
School: University of North...
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P Algebra art o f Qualifying Examination, A ugust 25, 2009 Instructions: Do all questions, justify your answers with the necessary proofs. All rings are associative (not necessarily commutative) with identity a nd all modules are left unitary modules. We denote by .z t he r ing of integers, a nd by IR, C t he fields of real a nd complex numbers, respectively. 1. I f M is a module over a ring R, set ann M = {r E R...

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P Algebra art o f Qualifying Examination, A ugust 25, 2009 Instructions: Do all questions, justify your answers with the necessary proofs. All rings are associative (not necessarily commutative) with identity a nd all modules are left unitary modules. We denote by .z t he r ing of integers, a nd by IR, C t he fields of real a nd complex numbers, respectively. 1. I f M is a module over a ring R, set ann M = {r E R I rm = 0 for all m E M}. I f I is a left ideal of R, d enote by t( 1) t he sum of all two-sided ideals of R c ontained in I . You may assume t hat a nn M is a two-sided ideal of R, a nd t (I) is t he largest two-sided ideal of R c ontained in I. (a) (3 points) I f I is a left ideal of R , prove t hat a nn R / I = t(1). (b) (3 points) Let l i, h be left ideals of R. I f t he R-modules R / II a nd R /I2 are isomorphic, prove t hat t(II) = t(I2). (c) (4 points) Let H be a two-sided ideal of R. Prove t hat t here exists a simple R-module 5 for which H ~ ann 5 , a nd if H is a maximal two-sided ideal, t hen H = a nn 5 . Hint: you may use t he fact t hat every left ideal is contained in a maximal left ideal. 2. Let R be a ring, let M be an R-module, a nd let N be a submodule of M. Assume t hat M = 51 Ell 52 Ell . .. Ell 5 n is an internal direct sum where, for all i, S, is a simple submodule of M. For all i, set M; = Ell 5 j so t hat M = S, EB M;' iii n (a) (3 points) Prove t hat, for all i, M: is a maximal submodule of M, a nd .n M; = t =I O. (b) (3 points) I f N is a maximal submodule of M , prove t hat, for some N EB5j a nd N i, M= ~ Mj. (c) points) (4 I f N is a simple submodule of M, prove t hat, for some i, M a nd N ~ 5 j . Hint: examine N n s; = NEBMj N n Mj. 3. Consider t he r ing R = a ll, a 21 E C, a 22 E IR a nd multiplication. a I2 (~ ~) of all 2 x 2 matrices A = (aij) satisfying = 0, with the usual operations of m atrix a ddition and (a) (2 points) F ind t he c enter of R, Z(R) = { z E R I zr = r z for all r E R}. (b) (1 point) Is t he r ing R left artinian? (c) (1 points) Is t he ring R left noetherian? (d) (3 points) Find the radical of R, J(R), a nd describe t he ring s tructure of R / J(R) in terms of IR a nd C. (e) (3 points) Describe t he nonisomorphic simple left R-mod ules by indicating their underlying abelian group a nd R-action. 1 4. Consider t he e xact sequence 0 ----t Z ~ 2 ~ 2 /32 ----t 0 of 2-modules where /'i;( z) = 3z for all z E 2 . Using t hat functor Hom is left exact a nd functor 0 is right exact, determine whether t he following sequences of 2-modules are exact. (a) (10 points) T he sequence obtained by tensoring t he above sequence with 2 /32, o ----t 2 0 z 2 /32 lI:~d 2 0 z 2 /32 7r~d 2 /32 0 z 2 /32 ----t O. (b) (5 points) T he sequence obtained by tensoring t he above sequence with 2 , o ----t 2 0 z 2 lI:~d 2 0 z 2 7r~d 2 /32 0 z 2 ----t O. (c) (10 points) T he sequence obtained by homming t he above sequence into 2 , o ----t H omz(2/32, 2 ) Ho~,Z) Homz(2, 2 ) Ho~,Z) H omz(2, 2 ) ----t O. (d) (5 points) T he sequence obtained by homming 2 into t he above sequence, o ----t H omz(2, 2 ) Ho~,II:) Homz(2, 2 ) Ho~,7r) H omz(2, 2 /32) ----t O. 2
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University of North Carolina School of the Arts - MATH - 601
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Algebra Part of Qualifying Examination, August 23, 2010Instructions: Do all questions, justify your answers with the necessary proofs. Allrings are associative (not necessarily commutative) with identity, and all modules are leftunitary modules. We den
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University of North Carolina School of the Arts - MATH - 680
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Qualifying Exam in Combinatorics19 August, 20081. Let be a 3-connected planar graph with planar dual . Prove ordisprove:If is a Cayley graph and is is vertex-transitive, then is a Cayleygraph.If you believe the statement to be true, give a proof, or
University of North Carolina School of the Arts - MATH - 680
University of North Carolina School of the Arts - MATH - 680
University of North Carolina School of the Arts - MATH - 680
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University of North Carolina School of the Arts - MATH - 680
University of North Carolina School of the Arts - MATH - 680
University of North Carolina School of the Arts - MATH - 680
University of North Carolina School of the Arts - MATH - 680
University of North Carolina School of the Arts - MATH - 680
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University of North Carolina School of the Arts - MATH - 680
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University of North Carolina School of the Arts - MATH - 680
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University of North Carolina School of the Arts - MATH - 680
University of North Carolina School of the Arts - MATH - 680
University of North Carolina School of the Arts - MATH - 680
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University of North Carolina School of the Arts - MATH - 661
University of North Carolina School of the Arts - MATH - 661
University of North Carolina School of the Arts - MATH - 661
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University of North Carolina School of the Arts - MATH - 661
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