Math 343, Intro to Algebraic Structures, Spring 2010
Homework 10 Solutions
p. 174, #8 Consider Z+ with the usual addition and multiplication. Decide if Z+ is a
ring. If a ring is not formed, tell why this is the case. If a ring is formed,
state whether the ring is commutative, whether it has unity, and whether
it is a field.
The structure Z+ does not form a ring. There is no additive identity
element so hZ+ , +i is not a group.
p. 174, #18 Describe all units of the ring Z × Q × Z.
The only units in Z are −1 and 1. All nonzero elements of Q are units.
Therefore the set of units in Z × Q × Z is the set
{(a, r, b) ∈ Z × Q × Z | a, b ∈ {−1, 1}, r ∈ Q∗ }.
p. 174, #33 Mark each of the following as true or false.
b. Every ring has a multiplicative identity.
This statement is false. The ring 4Z has no multiplicative identity.
d. Every ring with unity has at most two units.
This statement is false. The ring Q has unity and infinitely many
units.
f. The distributive laws for a ring are not very important.
This statement is false. The distributive laws are very important in
that they link the two binary operations on the ring.
h. The nonzero elements of a field form a group under the multiplication
in the field.
This statement is true. It is easy to give a proof.
j. Every element in a ring has an additive inverse.
This statement is true. In a ring R, we know hR, +i is an abelian
group. In particular every element of R has an additive inverse.
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p. 182, #17 Mark each of the following as true or false.
b. Every field is an integral domain.
This statement is true. Let F be a field and suppose ab = 0 for some
a, b ∈ F , a 6= 0. Multiplying both sides of the equation by a−1 yields
b = 0. We conclude F has no zero divisors. Therefore F is an integral
domain.
d. As a ring, Z is isomorphic to nZ for all n ≥ 1.
This statement is false. We will prove Z is not isomorphic to 2Z.
Suppose φ : Z → 2Z is a ring isomorphism. We claim that φ(1) is a
multiplicative identity for 2Z. Let x ∈ 2Z. Since φ is onto there exists
a ∈ Z such that φ(a) = x. Now consider
φ(1)x = φ(1)φ(a)
= φ(1a)
= φ(a)
= x.
Since 2Z is a commutative ring we also have xφ(1) = x. Therefore
φ(1) is a multiplicative identity for 2Z. But this is a contradiction
since we know 2Z is not a unital ring.
f. Every integral domain of characteristic 0 is infinite.
This statement is true. Note that the set {n · 1 | n ∈ Z} will be an
infinite subset of any integral domain of characteristic 0.
h. A divisor of zero in a commutative ring with unity can have no multiplicative inverse.
This statement is true. Let R be a commutative ring with unity.
Suppose that a ∈ R is a zero divisor and has a multiplicative inverse,
say b ∈ R. Then a 6= 0 and there exists a nonzero c ∈ R such that
ac = 0. Multiplying the last equation by b yields c = 0, which is a
contradiction.
j. Z is a subfield of Q.
This statement is false. The ring Z is not a field.
p. 182, #23 An element of a ring R is idempotent if a2 = a. Show that a division ring
contains exactly two idempotent elements.
Proof. Let R be a division ring. In particular R has at least two idempotent
elements namely 0 and 1. Now suppose a ∈ R is idempotent. From the
equation a2 = a we get a2 − a = 0, that is a(a − 1) = 0. Since R has
no zero divisors we see that a = 0 or a = 1. Therefore R has exactly two
idempotents.