THE UNIVERSITY OF WESTERN ONTARIO
LONDON
CANADA
DEPARTMENT OF MATHEMATICS
Math 3020A Midterm Exam
October 22, 2010
3 hours
1. Let R denote the set of all positive real numbers. Dene two operations on R, called
addition and multiplication, respectively, as
Assignment 3, due at classtime, Thursday, November 25
1. Let R and S be rings, and f : R S be a homomorphism.
(a) Prove that if I is an ideal of S , then f 1 (I ) = cfw_ x R | f (x) I is an ideal of R.
Solution: Let I be an ideal of S . Since f (0) = 0 I
Assignment 1, due at classtime, Tuesday, October 5
1. Let a, b Z be such that not both are zero. Prove that not both a + b and ab are zero,
and that if ( a, b ) = 1, then ( a + b, ab ) = 1.
Solution: Suppose that one of a and b is zero. Since a + b and ab
Mathematics 3020A Assignment 7, due at classtime, Thursday, November 7
1. Let G, H be groups.
(a) Prove that G1 = cfw_ (g, eH ) | g G and H1 = cfw_ (eG , h) | h H are normal subgroups
of G H .
Solution: Let (g, h) G H . Then (g, h)1 = (g 1 , h1 ), so
(g
Mathematics 3020A Assignment 11, due at classtime, Thursday, December 5
1. Prove that if R is a commutative ring with identity, and I is an ideal of R for which R/I
is a eld, then I is maximal.
Solution: Let J be an ideal of R with I J . We must prove tha
Mathematics 3020A Assignment 10, due at classtime, Thursday, November 28
1. Let R be a commutative ring with identity, and dene D : R[x] R[x] as follows: let
f R[x], and dene D (f ) : N R by (D (f )(i) = (i+1)f (i+1) for i N. Recall that this
is the deriv
Mathematics 3020A Assignment 9, due at classtime, Thursday, November 21
1. Let D be an integral domain, and let FD = (D (D cfw_ 0 )/ , where is the equivalence
relation dened on D D cfw_ 0 by (a, b) (c, d) if and only if ad = bc. Let a denote
b
[(a, b)].
Mathematics 3020A Assignment 8, due at classtime, Thursday, November 14
1. Let R be a ring. Dene the centre of R, denoted by Z (R), by
Z (R) = cfw_ x R | for all y R, xy = yx .
(a) Prove that Z (R) is a subring of R.
Solution: Let a, b Z (R). Then for any