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### Algebra2010Aug

Course: MATH 601, Spring 2011
School: University of North...
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Word Count: 861

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Part Algebra of Qualifying Examination, August 23, 2010 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 R, Q the elds of real and rational numbers, respectively. 1. Given a ring R, the opposite ring Rop has the same underlying...

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Part Algebra of Qualifying Examination, August 23, 2010 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 R, Q the elds of real and rational numbers, respectively. 1. Given a ring R, the opposite ring Rop has the same underlying abelian group as R and a new multiplication dened by a b = ba, for all a, b R, where juxstaposition denotes the original multiplication in R. Let Mm,n (R) be the set of m n matrices with entries from R. The set Mn (R) = Mn,n (R) is a ring with respect to the usual addition and multiplication of matrices. The sets Mn (R) and Mn (Rop ) coincide as abelian groups, but have dierent ring structures. For all A Mm,n (R), a R, we write a A = Aa and A a = aA for the scalar multiplication. For all B Mn,p (R), we write AB for the product of A and B over R, and A B for their product over Rop . (a) Consider R as an R-module and prove the following. (i) (1 point) For any a R, the map a : R R given by a (r) = a r, for all r R, is an endomorphism of R (as an R-module). (ii) (2 points) For any EndR (R), there is a unique a R satisfying (r) = a r, for all r R. (iii) (2 points) The map Rop EndR (R) given by a a is an isomorphism of rings. (b) Consider L = Mn,1 (R) as an R-module and prove the following. (i) (1 point) For any A Mn (R), the map A : L L given by A (X ) = A X, for all X L, is an endomorphism of L. (ii) (2 points) For any EndR (L), there is a unique A Mn (R) satisfying (X ) = A X, for all X L. (iii) (2 points) The map Mn (Rop ) EndR (L) given by A A is an isomorphism of rings. (c) Consider L as an Mn (R)-module and prove the following. (i) (2 points) For any a R, the map a : L L given by a (X ) = a X, for all X L, is an endomorphism of L. (ii) (4 points) For any EndMn (R) (L), there is a unique a R satisfying (X ) = a X, for all X L. (iii) (2 points) The map Rop EndMn (R) (L) given by a a is an isomorphism of rings. (d) points) (2 Consider N = Mn,p (R) as an Mn (R)-module. State, but DO NOT prove the analogs of (i)(iii) of part (c). 2. Let R be an arbitrary ring. (a) (2 points) Give the denition of when an R-module P is projective. (b) (2 points) Do projective R-modules exist? You may quote an appropriate theorem. 1 Algebra Part of Qualifying Examination, August 23, 2010, Page 2 2. (continued) (c) (4 points) If P is a projective R-module, prove that an exact sequence of R-modules of the form 0 X Y P 0 splits. Begin with a denition of a split exact sequence. (d) (5 points) Prove that an R-module A is projective if and only if for each exact sequence 0 X Y Z 0 of R-modules, the sequence 0 HomR (A, X ) HomR (A,) HomR (A, Y ) HomR (A, ) HomR (A, Z ) 0 of abelian groups is exact. You may use the denition of a projective module and the left exactness of the functor Hom. 3. Let R be an arbitrary ring. (a) (2 points) Give the denition of when an R-module I is injective. (b) (2 points) State Baers criterion for when an R-module I is injective. (c) (6 points) Using Baers criterion, prove that if R is a (commutative) principal ideal domain (PID), then an R-module A is injective if and only if it is divisible. First give a denition of a divisible module. (d) (2 points) Do divisible modules over a PID exist? Explain. (e) (2 points) Explain how one can construct an injective module over an arbitrary ring R using a divisible Z-module. Quote an appropriate statement. R0 of all 2 2 matrices A = (aij )1i,j 2 satisfying RQ Q, and a12 = 0, with the usual operations of matrix addition and 4. Consider the ring R = a11 , a21 R, a22 multiplication. (a) (1 points) Is the ring R left artinian? (b) (1 points) Is the ring R left noetherian? (c) (1 points) Is the ring R right artinian? (d) (1 points) Is the ring R right noetherian? (e) (5 points) Find the radical of R, J (R), and describe the ring structure of R/J (R) in terms of R and Q. (f) (4 points) Describe the nonisomorphic simple left R-modules by indicating their underlying abelian group and R-action. 2
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