Spring 2017

Math 5120 Graduate Algebra II

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Homework 2

Due Friday March 10, in class
  • Section 14.6
    • Homework: 2, 6, 7, 13, 17, 18, 40 (use the suggested procedure or solve by other means)
    • Hand In: 2(c), 6, 13, 18, 40
  • Section 14.7
    • Homework: 1, 2, 10
    • Hand In: 2
  • Section 14.8
    • Homework: 3, 4, 6, 7
    • Hand In: 4, 6
  • Section 10.1
    • Homework: 3, 4, 8, 13, 21 and the following problems:
      • 24. Suppose that \(R\) is a commutative ring with unity. A (left) module is irreducible or simple if it is not the zero module and if it has no proper submodule.
        1. Prove that any irreducible module is isomorphic to \(R/M\), where \(M\) is a maximal ideal of \(R\).
        2. Let \(\varphi : S \to S'\) be a homomorphism of irreducible \(R\)-modules. Prove that either \(\varphi\) is the zero map or that \(\varphi\) is an isomorphism.
        (Source: Michael Artin, "Algebra", Prentice Hall, Englewood Cliffs, NJ (1991))
      • 25. The annihilator of a (left) \(R\)-module \(V\) is the set \(I = \{r \in R \vert rV = 0\}\).
        1. Prove that \(I\) is a (2-sided) ideal of \(R\)
        2. Find the annihilator of the \({\mathbb Z}\)-module \({\mathbb Z}/2{\mathbb Z} \times {\mathbb Z}/3{\mathbb Z}\times {\mathbb Z}/4{\mathbb Z}\).
        (Source: Michael Artin, "Algebra", Prentice Hall, Englewood Cliffs, NJ (1991))
    • Hand In: 24, 25
  • Section 10.2
    • Homework: 3, 4, 5, 6, 9, 10 and the following problem:
      • 15. Prove that the ring of endomorphisms of an irreducible module over a commutative ring with unity is a division ring. (Source: Michael Artin, "Algebra", Prentice Hall, Englewood Cliffs, NJ (1991))
    • Hand In: 15
  • Section 10.3
    • Homework: 1, 4, 9, 10, 11 and the following problem:
      • 28. Let \(R\) be a commutative ring with unity. Let \(I\) be a nonzero ideal of \(R\). Prove that \(I\) is a free \(R\)-module if and only if it is a principal ideal of \(R\) and is generated by an element that is not a zero divisor in \(R\). (Source: Michael Artin, "Algebra", Prentice Hall, Englewood Cliffs, NJ (1991))
    • Hand In: 28