Algebra2008Jan - Qualifying Examination Algebra Part •...

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Unformatted text preview: Qualifying Examination January 10, 2008 Algebra Part • Please do all ﬁve questions. • Problem #5 is worth twice as much as each of the others We will always assume that rings have an identity element and that modules are unitary left modules. 1. Let I be an ideal of a ring R and M an R-module. Prove that there is an isomorphism of left R-modules HomR (R/I, M ) ∼ {m ∈ M |Im = 0 } = (You may assume that the subset {m ∈ M |Im = 0 } is an R-submodule of M .) 2. Prove directly from the deﬁnition that an R-module P is projective if and only if HomR (P, −) is exact. That is, prove that applying HomR (P, −) to any short exact sequence produces a short exact sequence. (You may assume that Hom is always half-exact.) 3. Use exact sequences to show that for any integers m, n > 0, one has an isomorphism Z/mZ ⊗Z Z/nZ ∼ Z/(m, n) = (You do not need to show that the ideal (m, n) is the principal ideal (gcd(m, n)).) 4. Prove that an Artinian ring is isomorphic to a direct product of ﬁnitely many division rings if and only if it contains no nonzero nilpotent element. Hint: J (R). 5. Justify your answers completely in each part below. a) Is Z/6Z a semisimple Z-module? b) Give a short exact sequence of Z-modules in which the outside terms are both semisimple modules, but the middle term is not semisimple. c) Is Z/2Z a projective Z/6Z-module (via the obvious structure)? d) Does Z have ﬁnite length (as a Z-module)? ...
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