Download Sample Exam #1

Math 0430
Exam #1 (Solutions)
05-3
1. (40) pts. Let a, b, d, p, n   with b 0 and n > 1. Let <R, +, > and <S, , > be rings.
Define or tell what is meant by the following:
(a) b divides a (b| a)
(b) d is the greatest common divisor of a and b (d = (a,b))
(c) p is prime
(d) a and b are relatively prime
(e) a is congruent to b modulo n ( a  b (mod n))
(f) z is a zero divisor of the ring R.
(g) <R, +, > is a division ring.
(h) f: R  S is a homomorphism.
2. (5) State the Fundamental Theorem of Arithmetic.
3. (15) State the Well-Ordering Principle or Axiom. Use it to prove one of the following:
(a) Let a and b be integers with b> 0. Then there exist integers q and r such that
a = qb + r and 0  r < b.
(b) Let a and b be integers, not both 0 and let d be their greatest common divisor.
Then there exist integers u and v such that d = au + bv and d is the smallest positive
integer that can be expressed in the form au + bv.
(c) Every integer n > 1 can be written as a product of primes.
4. (10) Use the Euclidean Algorithm to find (91,105) and express it as a linear combination
of 91 and 105.
Proof: 105 = (1)91 + 14
91 = (6)14+ 7
14 = (2)7 + 0
(91,105) = 7
7 = 91 – (6)14 = 91 – 6(105 – 91) = (7)91 – (6)105
5. (10) Prove that if a | bc and a and b are relatively prime, then a | c.
Proof: If a | bc and a and b are relatively prime, then there are integers x, y, and u
such that bc = au and 1 = ax + by. c = c(1) = c(ax + by) = cax + cby = cax + auy =
a (cx + uy). So a | c.
6. (15) Prove that Zn is an integral domain if and only if n is prime.
Proof: If n is not prime, there are integers a and b such that n = ab with 1<a<n and 1<b<n.
Then ab = 0 in Zn , but a ≠0 and b ≠0. This means that Zn is not an integral domain.
Conversely, suppose that n is prime. Let a be a nonzero element of Zn with ax = 0 for some
x in Zn. Since a is a nonzero elelment of Zn, n does not divide a and (a,n) =1. So there are
integers r and t such that 1= ra + tn. Therefore, x = rax + tnx = 0. Since Zn has no zero
divisors, it is an integral domain.
7. (10) Prove that If n is an odd integer, n2  1 (mod 8).
Proof: Since n is odd, it is of the form 4k + 1 or 4k + 3. So n2 is of the form 16k2 + 8k + 1 =
8(2k2 + k) + 1 or 16k2 + 24 k + 9 = 8(2k2 +3K + 1) + 1. In either case, 8| n2 -1 and
n2  1 (mod 8).
a 0
 | a, b  R} is a subring of M(R).
8. (10) Show that S = { 
0 b
0 0
a 0  c 0
a 0  c 0
  S, so S is nonempty. Let 
 , 
  S. Then 
 - 
 =
Proof: 
0 0
 0 b   0 d
 0 b   0 d
0 
a  c
 a 0   c 0   ac 0 

  S and 
  S. This is sufficient to show that S is a
 · 
 = 
b  d
 0
 0 b   0 d   0 bd
subring of M(R).
9. (15) Let operations of “addition” and “multiplication” be defined on  by a b = a + b – 1
and ab = ab – (a + b) +2. Prove that , ,  is isomorphic to , +, , the ring of
integers with ordinary addition and multiplication.
Proof: Define f: , ,   , +,  by f(a) = a - 1. If a, b  , ,  are such that
f(a) = f(b), then a - 1 = b - 1 and a = b. Therefore, f is injective. Let z  , +, . Then
(z +1)  , ,  and f(z +1) = (z + 1) – 1 = z. Therefore, f is surjective.
If a, b  , , , then f(ab) = f(a + b – 1) = (a + b – 1) - 1= (a – 1) + (b – 1) =
f(a)+ f(b). Also f(ab) = f(ab – (a + b) +2) = ab – (a + b) +2 - 1 = ab – a – b + 1 =
(a – 1)(b – 1) = f(a)·f(b). Since f preserves the operations of addition and multiplication, it is
a homomorphism. Hence, , ,   , +, .
You may earn up to 10 points extra credit by proving that a field is an integral domain.
Let R be a field. Then R is a commutative ring with an identity 1R ≠0R. Let a,b  R with
ab=0R. Suppose that a ≠ 0R. Then (since R is a field) a is a unit and there is an element
u  R such that ua = 1R. So b = 1R b = (ua)b = u(ab) = u0R = 0R and R is an integral
domain.
10. (20) Let T = {w, x, y, z} be the following ring of order 4:
T

w
x
y
z

w
x
y
z
w
w
x
y
z
w
w
w
w
w
x
x
y
z
w
x
w
y
w
y
y
y
z
w
x
y
w
w
w
w
z
z
w
x
y
z
w
y
w
y
The characteristic of a ring R is the least positive integer m such that ma = 0R for every aR.
If no such integer exists, the ring is said to have characteristic 0.
An element e of a ring R is said to be idempotent if e2 = e.
(a) Complete the multiplication table for T.
(b) Find 3x - xz. z-y = z + y = x
(c) Find x2z2. yy=w
(d) Find the characteristic of T. 4
(e) Find the zero divisors of T. x, y, and z
(f) Find the idempotent elements of T. w
The ring of integers with ordinary addition and multiplication is a ring of characteristic zero.