Download Chapter 11:

Chapter 11:
1) What is the smallest positive integer n such that there are two nonisomorphic groups of
order n?
2) What is the smallest positive integer n such that there are three nonisomorphic Abelian
groups of order n?
3) What is the smallest positive integer n such that there are exactly four nonisomorphic
Abelian groups of order n?
4) Calculate the number of elements of order 2 in each of Z16, Z8
Z4 Z2 Z2. Do the same for the elements of order 4.
Z2, Z4
Z4, and
5) Prove that any Abelian group of order 45 has an element of order 15. Does every
Abelian group of order 45 have an element of order 9?
9) Suppose that G is an Abelian group of order 120 and that G has exactly tree elements
of order 2. Determine the isomorphism class of G.
12) Suppose that the order of some finite Abelian group is divisible by 10. Prove that the
group has a cyclic subgroup of order 10
19) The set {1, 9, 16, 22, 29, 53, 74, 79, 81} is a group under multiplication modulo 91.
Determine the isomorphism class of this group.
20) Suppose that G is a finite Abelian group that has exactly one subgroup for each
divisor of |G|. Show that G is cyclic.
Chapter 12:
14) Let a and b belong to a ring R and let m be an integer. Prove that m *(ab) = (m*a)b =
a(m*b).
18) Let a belong to a ring R. Let S = {x Є R | ax = 0}. Show that S is a subring of R
19) Let R be a ring. The center of R is the set {x Є R | ax = xa for all a in R}. Prove that
the center of a ring is a subring.
20) Describe the elements of M2(Z) (see example 4) that have multiplicative inverses.
23) Determine U(Z[i]) (See example 11).
30) Suppose that there is an integer n > 1 such that xn = x for all elements x of some ring.
If m is a positive integer and am = 0 for some a, show that a = 0
32) Let n be an integer greater than 1. In a ring in which xn = x for all x, show that ab = 0
implies ba = 0.
Chapter 13:
3) Show that a commutative ring with cancellation property (under multiplication) has no
zero-divisors
4) List all zero-divisors in Z20. Can you see a relationship between the zero-divisors of
Z20 and the units of Z20?
5) Show that every nonzero element of Zn is a unit or a zero-divisor.
6) Find a nonzero element in a ring that is neither a zero-divisor nor a unit.
12) Find two elements a and b in a ring such that both a and b are zero-divisors, a + b ≠ 0
and a + b is not a zero-divisor.
13) Let a belong to a ring R with unity and suppose that an = 0 for some positive integer
n. (such an element is called nilpotent.) Prove that 1 – a has a multiplicative inverse in R.
[Hint: Consider (1 - a) (1 + a + a2 + … + an-1).]
14) Show that the nilpotent elements of a commutative ring form a subring.
25) Let R be a ring with unity 1. If the product of any pair of nonzero elements of R is
nonzero, prove that ab = 1 implies ba = 1
50) Find the characteristic of Z4
4Z
52) In a commutative ring of characteristic 2, prove that the idempotents form a subring.
58) Suppose that F is a field with characteristic not 2, and that the nonzero element of F
form a cyclic group under multiplication. Prove that F is finite.
59) Suppose that D is an integral domain and that Φ is a nonconstant function from D to
the nonnegative integers such that Φ(xy) = Φ(x) Φ(y). If x is a unit in D, show that
Φ(x) = 1.