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1、NATIONAL UNIVERSITY OF SINGAPORE FACULTY OF SCIENCE SEMESTER 2 EXAMINATION 2006-2007 MA3201Algebra II 30 April 2007 Time allowed : 2 hours INSTRUCTIONS TO CANDIDATES 1. This examination paper contains FIVE (5) questions and comprises THREE (3) printed pages. 2. Answer ALL questions. 3. Candidates ma
2、y use calculators.However, they should lay out systematically the various steps in the calculations. PAGE 2MA3201 Answer all questions. Question 120 marks (a) Let f : R S be a function between two rings R and S. (i) Explain what is meant to say that f is a ring homomorphism. (ii) When f is a ring ho
3、momorphism, defi ne its kernel, denoted as ker(f). (b) Let R, S and T be rings, and suppose that : R S and : R T are ring homomorphisms. (i) Show that, if there exists a ring homomorphism : S T such that = , then ker() ker(). (ii) Prove that, if ker() ker() and is surjective, then there exists a uni
4、que ring homomorphism : S T such that = . Question 220 marks (a) Let I be an ideal of a commutative ring with 1. Explain briefl y what is meant to say that (i) I is prime; (ii) I is maximal. (b) Let a be an element in an integral domain. Explain briefl y what is meant to say that (i) a is prime; (ii
5、) a is irreducible. (c) Let R be a unique factorisation domain, and F be its fi eld of fractions. Let f(X) RX be an irreducible element of RX such that deg(f(X) 1, and denote the principal ideals it generates in RX and FX by I and J respectively, i.e. I = f(X)a(X) | a(X) RX; J = f(X)b(X) | b(X) FX.
6、(i) Show that I is prime in RX, and J is maximal in FX. (ii) Provide an example to show that I may not be maximal. . - 3 - PAGE 3MA3201 Question 320 marks (a) Explain what is meant by (i) a Euclidean function, and (ii) a Euclidean domain. (b) Prove or disprove the following statement: A fi eld is a
7、Euclidean domain. (c) Prove that Z2 is a Euclidean domain. (You may assume that the function N : Q 2 Q defi ned by N(a + b2) = a2 2b2is multiplicative without proof.) Question 420 marks Let R be a ring and 1, and let M be an R-module. (a) Suppose that N1,N2,.,Nrare submodules of M. Explain what is m
8、eant to say that the sum N1+ N2+ + Nris direct. (b) Suppose that there exist submodules U1,U2,.,Urof M such that M = U1 U2 Ur. For each i = 1,2,.,r, let Vibe a submodule of Ui. (i) Prove that the sum V1+ V2+ + Vris direct. (ii) Let V = V1V2Vr. Prove that M/V = (U1/V1)(U2/V2)(Ur/Vr). Question 520 mar
9、ks Explain what is meant by a zero divisor of a ring. The characteristic of a ring with 1 is the least positive integer n such that n times z| 1 + 1 + + 1 = 0 if such an integer exists; if 1 + 1 + + 1 |z m times 6= 0 for all m Z+, then the characteristic of the ring is 0. Let R be a ring with 1 with
10、 characteristic n. (a) If R has no zero divisor, prove that either n = 0 or n is a prime integer. (b) Let S be another ring with 1, with characteristic m, and let : R S be a unital ring homomorphism. (i) Prove that there exists a Z such that ma = n. (ii) Deduce that if R has no zero divisor, and n 0, then m = n. END OF PAPER
