# Algebra2009Aug - Algebra P art o f Qualifying Examination,...

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Algebra Part of Qualifying Examination, August 25, 2009 Instructions: Do all questions, justify your answers with the necessary proofs. All rings are associative (not necessarily commutative) with identity and all modules are left unitary modules. We denote by .z the ring of integers, and by IR, C the fields of real and complex numbers, respectively. 1. If M is a module over a ring R, set ann M = {r E R I rm = 0 for all m EM}. If I is a left ideal of R, denote by t( 1) the sum of all two-sided ideals of R contained in I. You may assume that ann M is a two-sided ideal of R, and t(I) is the largest two-sided ideal of R contained in I. (a) (3 points) If I is a left ideal of R, prove that ann R/I = t(1). (b) (3 points) Let li, h be left ideals of R. If the R-modules R/ II and R/I 2 are isomorphic, prove that t(II) = t(I2). (c) (4 points) Let H be a two-sided ideal of R. Prove that there exists a simple R-module 5 for which H ~ ann 5, and if H is a maximal two-sided ideal, then H = ann 5. Hint:

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## This note was uploaded on 06/19/2011 for the course MATH 601 taught by Professor Na during the Spring '11 term at University of North Carolina School of the Arts.

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Algebra2009Aug - Algebra P art o f Qualifying Examination,...

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