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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 Rmodule. Prove that there is an isomorphism
of left Rmodules
HomR (R/I, M ) ∼ {m ∈ M Im = 0 }
=
(You may assume that the subset {m ∈ M Im = 0 } is an Rsubmodule of M .)
2. Prove directly from the deﬁnition that an Rmodule 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 halfexact.)
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 Zmodule?
b) Give a short exact sequence of Zmodules in which the outside terms
are both semisimple modules, but the middle term is not semisimple.
c) Is Z/2Z a projective Z/6Zmodule (via the obvious structure)?
d) Does Z have ﬁnite length (as a Zmodule)? ...
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 Spring '11
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 Algebra

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