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:
█
Example 7: Describe all units of the ring [pic].
Solution:
█
Fact: For [pic], [pic] is a unit only when [pic]
Example 8: Find all of the units for the ring [pic].
Solution:
█
Example 9: Find all of the units for the ring [pic].
Solution:
█
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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