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Rings and Fields

HW p.10 # 1-9 at the end of the notes

In this section, we discuss the basics of rings and fields.

Rings

Definition 1: A ring is a non-empty set R with two binary operations + and , normally called addition and multiplication, defined on R where the following axioms are satisfied for all :

1. : is an abelian group, that is

a.For , (R is closed under +)

b. (Associatively under + is satisfied)

c. For each , there exists an identity were

(R has an additive Identity)

d. For each , there exists an where

(Each element in R has an additive inverse)

e. (Addition is commutative)

2.: For , (R is closed under )

3.: (Associativity under is satisfied)

4.: (Left and Right Distributive laws are satisfied)

Definition2: A commutativering is a ring R that satisfies for all (it is commutative under multiplication). Note that rings are always by condition 1 commutative under addition.

Definition3: A ringwithunity is a ring with the multiplicative identity, that is, there exists where for all .


Examples of Rings

Example 1: Show that the integers represents a ring.

Solution: The integers represents a ring. For , we show it satisfies the 4 properties for a ring.

1. : is known to be an an abelian group, that is

a. For , it is known that Z is closed under + , that is

b. (Z is known to be associative under +)

c. For each , there exists an identity zero given by were

(0 is the known additive identity element in the integers)

d. For each , there exists an where

(Each element in Z has an additive inverse obtained by

negating the element)

e. (Z is known to be commutative under +)

2.: For , it is known that Z is closed under , that is

3.: (Z is known to be associative under )

4.: (Left and Right Distributive laws are known to hold in the

integers.)

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Notes:

i. Z is a commutative ring since the integers are known to be commutative under multiplication, that is for all .

ii.Z has unity 1 since for all .

Example 2: Show that is a ring. Is a commutative ring? Does it have unity?

Solution:

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Definition 4: The Cartesian product of the groups is the set , where for . We denote the Cartesian product by

.

Recall that a group G is a non-empty set that is closed under a binary operation * that satisfy the following 3 axioms

  1. Associativity: For all ,
  2. Identity: For any , there exists an where .
  3. Inverse: For each , there exists an element where .

Fact: The Cartesian product forms a group under the binary operation .

Proof:Note that G is closed. This is true because, since each is a group, each is closed and for any . Hence

since

We next prove the 3 group properties.

1. Associativity: Let . Then ,

, and where . It can be show that both and equal . Hence, is associative.

2.Identity: The identity is given by , where each is the identity for the group . Note that for, we have

.

Similarly, we can show .

3.Inverse. For each , since is a group. Hence, the inverse of is . Note that

Similarly, .

Hence, by definition, is a group.█

Example 3:Show is a ring under addition and multiplication.

Solution:Let . Then , , and where

.

We now demonstrate that this set satisfies the 4 properties for a ring,

: is an abelian group under + since

i) is closed under + since

ii) is associative under + since

iii) 0 = (0, 0) serves as the identity under + since

iv) For , then serves as the additive inverse since

v) is abelian under + since

is closed under since

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is associative under multiplication.

: The distributive laws hold. For example,

A similar argument can be used to show

Since all of the properties hold, is a ring.

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Example 4: Compute (-4, 7) (2, 8) in .

Solution:

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Note: The set of matrices with entries in a ring R is an example of a non-commutative ring since matrix multiplication is known not to be commutative.

Theorem 1: If R is a ring with additive identity of 0, then for any , we have

1. .

2.

3.

Proof:

1.

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2. We show that .

Now, .

Then, adding to both sides gives

Similarly, it can be shown that .

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3.Using property 2, we can show that

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Units

Definition5: Let R be a ring with unity . An element is a unit of R if it has a multiplicative inverse in R. That is, for , there exists an element where . If everynon-zeroelement of R is a unit, then R is a division ring. A field is a commutative division ring.

ExamplesofFields

The real numbers and rational numbers under the operations of addition + and multiplication are fields. However, the integers Z under addition + and multiplication

is not a field since the only non-zero elements that are units is -1 and 1. For example, the integer 2 has no multiplicative inverse since .

Example 5: Describe all units of the ring Z.

Solution:

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Example 6: Describe all units of the ring .

Solution:

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Example 7: Describe all units of the ring .

Solution:

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Fact: For , is a unit only when

Example 8: Find all of the units for the ring .

Solution:

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Example 9: Find all of the units for the ring .

Solution:

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Note: If p is a prime, then is a field since all non-zero elements are units.


Exercises

1.Determine if the following sets under the usual operations of addition and multiplication represent that of a ring. If it is a ring, state whether the ring is commutative, whether it has a unity element, and whether it is a field. If it is not a ring, indicate why it is not.

a. under usual addition and multiplication.

b. under usual addition and multiplication.

c. under usual addition and multiplication by components.

d.T he set of invertible matrices with real entries under usual addition and multiplication.

e. under usual addition and multiplication by components.

f. under usual subtraction and multiplication.

2. Compute the following products in the given ring.

a.(10)(8) in

b.(8)(5) in

c.(-10)(4) in

d.(2, 3)(3, 5) in

e.(-5, 3)(4, -7) in

3.Describe the units of the given rings.

  1. Z

4.Show that for all x, y in a ring R if and only if R is commutative.

5.Let be an abelian group. Show that is a ring if we define for all .

6.Show for the ring , that the expansion is true.

7.Show for the ring , where p is prime, that the expansion is true.

Hint: Note that for a commutative ring, the binomial expansion

,

where , is true.

8.Show that the multiplicative inverse of a unit in a ring with unity is unique.

9.An element of a ring R is idempotent if .

a.Show that the set of all idempotent elements of a commutative ring is closed under

multiplication.

  1. Find all idempotents in the ring .
  2. Show that if A is an matrix such that is invertible, then the matrix is an idempotent in the ring of matrices.