The University of British Columbia
Practice Final Examination - April, 2016
MATH 323
Closed book examination
Time: 2.5 hours
Last Name
First
Signature
Student Number
Special Instructions:
No memory aids are allowed. No calculators may be used. Show all your work; little or no
credit will be given for a numerical answer without the correct accompanying work. If you
need more space than the space provided, use the back of the previous page. Where boxes
are provided for answers, put your final answers in them. Information on this page must
be filled out in blue or black pen.
Rules governing examinations
• Each candidate must be prepared to produce, upon request, a
UBCcard for identification.
• Candidates are not permitted to ask questions of the invigilators,
except in cases of supposed errors or ambiguities in examination
questions.
• No candidate shall be permitted to enter the examination room
after the expiration of one-half hour from the scheduled starting
time, or to leave during the first half hour of the examination.
• Candidates suspected of any of the following, or similar, dishonest practices shall be immediately dismissed from the examination
and shall be liable to disciplinary action.
(a) Having at the place of writing any books, papers
or memoranda, calculators, computers, sound or image players/recorders/transmitters (including telephones), or other memory aid devices, other than those authorized by the examiners.
(b) Speaking or communicating with other candidates.
(c) Purposely exposing written papers to the view of other candidates or imaging devices. The plea of accident or forgetfulness
shall not be received.
• Candidates must not destroy or mutilate any examination material; must hand in all examination papers; and must not take any
examination material from the examination room without permission of the invigilator.
• Candidates must follow any additional examination rules or directions communicated by the instructor or invigilator.
Page 1 of 11 pages
1
36
2
10
3
12
4
17
5
17
6
8
Total
100
April, 2016
Math 323
Name:
Page 2 of 11 pages
[36] 1. Short Answer. Each question below is worth 3 marks and some require explanation/computation to obtain any marks. You may not be awarded points if your answer
is correct but your explanation is wrong. Please write your final solutions in the box if
provided.
√
(a) TRUE or FALSE: Z[ −3] is a PID. Justify.
Answer
(b) TRUE or FALSE: Any prime ideal of Z[x] is maximal. Justify.
Answer
(c) TRUE or FALSE: Any maximal ideal of Z[x] is prime. Justify.
Answer
April, 2016
Math 323
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(d) TRUE or FALSE: Every non-zero homomorphism of integral domains f : R → S must
satisfy f (1R ) = 1S . Justify.
Answer
(e) TRUE or FALSE: Every finitely generated module over Z[i] is a sum of a free module
and a torsion module. Justify.
Answer
(f) Order by inclusion the following special classes of rings: PIDs, fields, integral domains,
Euclidean domains, UFDs. For each inclusion, single out a concrete example showing
that inclusions are strict. (You may give answer in form of nested Venn diagrams.)
April, 2016
Math 323
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Page 4 of 11 pages
√
√
(g) TRUE or FALSE: Z[ 3] and Z[ 5] are isomorphic rings. Justify.
Answer
(h) TRUE or FALSE: For any two ideals I, J of an arbitrary ring R, one has R/(IJ) ∼
=
R/I × R/J. Justify.
Answer
(i) TRUE or FALSE: The polynomial f (x) = x7 + 15x5 − 75x4 + 20x − 10 is irreducible in
Z[x]. Justify.
Answer
April, 2016
Math 323
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(j) TRUE or FALSE: A direct sum of torsion modules (over an integral domain) is again a
torsion module. Justify.
Answer
(k) TRUE or FALSE: A quotient of a torsion module is always a torsion module. Justify.
Answer
(l) TRUE or FALSE: A quotient of a free module is always free. Justify.
Answer
April, 2016
Math 323
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[10] 2. Let p and q be distinct primes, and let m be an arbitrary integer. Compute
HomZ (Z/pZ × Z/qZ, Z/mZ).
April, 2016
Math 323
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[12] 3. Prove or disprove:
(a) (4 points) The rings Z[x]/(x2 + 1) and Z[x]/(x2 + 2) are isomorphic.
(b) (4 points) The Z-modules Z[x]/(x2 + 1) and Z[x]/(x2 + 2) are isomorphic.
(c) (4 points) The Z[x]-modules Z[x]/(x2 + 1) and Z[x]/(x2 + 2) are isomorphic.
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[17] 4.
√
(a) (12 points) Prove or disprove: is 5 a prime in Z[ 1+ 2−19 ]? If not, factor it as a product
of primes, with proof that the factors are prime.
√
(b) (5 points) Find an example of an element of Z[ −5] that is irreducible but not prime
(and give a complete proof that it has this property).
April, 2016
Math 323
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[17] 5.
(a) (6 points) Let R = Z[x] and M = (2, x) be the ideal generated by 2 and x, considered
as a submodule of R. Show that {2, x} is not a basis of R. Show that rank of M is 1
but it is not free of rank 1.
(b) (4 points) Show that if R is an integral domain and M is any non-principal ideal of R
then M is torsion free of rank 1 but it is not a free R-module.
√
(c) (7 points) Prove or disprove: Any finitely generated module over Z[ −5] is sum of a
free module and a torsion module.
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[8] 6. Count the number of finitely generated abelian groups of Z-rank less or equal 5 and
whose torsion has order 100,000.
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Math 323
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will not be graded