Section 1: Rings and Fields - Radford



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

1. [pic]: [pic] is an abelian group, that is

a. For [pic], [pic] (R is closed under +)

b. [pic] (Associatively under + is satisfied)

c. For each [pic], there exists an identity [pic] were

[pic] (R has an additive Identity)

d. For each [pic], there exists an [pic] where

[pic] (Each element in R has an additive inverse)

e. [pic] (Addition is commutative)

2. [pic]: For [pic], [pic] (R is closed under [pic])

3. [pic]: [pic] (Associativity under [pic] is satisfied)

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

[pic]

Definition 2: A commutative ring is a ring R that satisfies [pic] for all [pic] (it is commutative under multiplication). Note that rings are always by condition 1 commutative under addition.

Definition 3: A ring with unity is a ring with the multiplicative identity, that is, there exists [pic] where [pic] for all [pic].

Examples of Rings

Example 1: Show that the integers [pic] represents a ring.

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

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

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

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

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

[pic] (0 is the known additive identity element in the integers)

d. For each [pic], there exists an [pic] where

[pic] (Each element in Z has an additive inverse obtained by

negating the element)

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

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

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

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

[pic] integers.)

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Notes:

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

ii. Z has unity 1 since [pic] for all [pic].

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

Solution:

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

[pic].

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 [pic], [pic]

2. Identity: For any [pic], there exists an [pic] where [pic].

3. Inverse: For each [pic], there exists an element [pic] where [pic].

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

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

[pic] since [pic]

We next prove the 3 group properties.

1. Associativity: Let [pic]. Then [pic],

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

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

[pic].

Similarly, we can show [pic].

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

[pic]

Similarly, [pic].

Hence, by definition, [pic] is a group. █

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

Solution: Let [pic]. Then [pic], [pic], and [pic] where

[pic].

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

[pic]: [pic]is an abelian group under + since

i) [pic]is closed under + since

[pic]

ii) [pic] is associative under + since

[pic]

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

[pic]

iv) For [pic], then [pic][pic] serves as the additive inverse since

[pic]

v) [pic] is abelian under + since

[pic]

[pic] [pic]is closed under [pic]since

[pic]

Continued on Next Page

[pic] [pic] is associative under multiplication.

[pic][pic]

[pic]: The distributive laws hold. For example,

[pic]

A similar argument can be used to show [pic]

Since all of the properties hold, [pic]is a ring.

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

Solution:

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Note: The set of [pic] 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 [pic], we have

1. [pic].

2. [pic]

3. [pic]

Proof:

1.

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2. We show that [pic].

Now, [pic].

Then, adding [pic] to both sides gives

[pic]

Similarly, it can be shown that [pic].

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3. Using property 2, we can show that

[pic]

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Units

Definition 5: Let R be a ring with unity [pic]. An element [pic] is a unit of R if it has a multiplicative inverse in R. That is, for [pic], there exists an element [pic] where [pic]. If every non-zero element of R is a unit, then R is a division ring. A field is a commutative division ring.

Examples of Fields

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

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 [pic].

Example 5: Describe all units of the ring Z.

Solution:

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Example 6: Describe all units of the ring [pic].

Solution:

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Example 7: Describe all units of the ring [pic].

Solution:

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Fact: For [pic], [pic] is a unit only when [pic]

Example 8: Find all of the units for the ring [pic].

Solution:

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Example 9: Find all of the units for the ring [pic].

Solution:

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Note: If p is a prime, then [pic] 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. [pic] under usual addition and multiplication.

b. [pic] under usual addition and multiplication.

c. [pic] under usual addition and multiplication by components.

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

e. [pic] under usual addition and multiplication by components.

f. [pic] under usual subtraction and multiplication.

2. Compute the following products in the given ring.

a. (10)(8) in [pic]

b. (8)(5) in [pic]

c. (-10)(4) in [pic]

d. (2, 3)(3, 5) in [pic]

e. (-5, 3)(4, -7) in [pic]

3. Describe the units of the given rings.

a. Z

b. [pic]

c. [pic]

d. [pic]

e. [pic]

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

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

6. Show for the ring [pic], that the expansion [pic] is true.

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

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

[pic],

where [pic], 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 [pic].

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

multiplication.

b. Find all idempotents in the ring [pic].

c. Show that if A is an [pic] matrix such that [pic] is invertible, then the [pic] matrix [pic] is an idempotent in the ring of [pic] matrices.

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