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Questions and Answers
What is a nonempty set R that has two closed binary operations, addition and multiplication?
What is a nonempty set R that has two closed binary operations, addition and multiplication?
Ring
What are the six conditions that satisfy a Ring?
What are the six conditions that satisfy a Ring?
1.a + b = b + a for a, b ∈ R. 2.(a + b) + c = a + (b + c) for a, b, c ∈ R. 3.There is an element 0 in R such that a + 0 = a for all a ∈ R. 4.For every element a ∈ R, there exists an element −a in R such that a + (−a) = 0. 5.(ab)c = a(bc) for a, b, c ∈ R. 6.For a, b, c ∈ R, a(b+c) = ab + ac and (a+b)c = ac + bc.
If there is an element 1 ∈ R such that 1 ≠0 and 1(a) = a(1) = a for each element a ∈ R, we say that R is a ring with what?
If there is an element 1 ∈ R such that 1 ≠0 and 1(a) = a(1) = a for each element a ∈ R, we say that R is a ring with what?
Multiplicative Identity
A ring R for which ab = ba for all a, b in R is called what?
A ring R for which ab = ba for all a, b in R is called what?
What is an Integral Domain?
What is an Integral Domain?
What is a Division Ring?
What is a Division Ring?
What property of a Division Ring is not always required?
What property of a Division Ring is not always required?
What is a Field?
What is a Field?
What is a Zero Divisor?
What is a Zero Divisor?
What is a Subring?
What is a Subring?
When is a subset S of R considered a subring?
When is a subset S of R considered a subring?
If R is a ring and r is a nonzero element in R, then r is said to be what if there is some nonzero element s ∈ R such that rs = 0?
If R is a ring and r is a nonzero element in R, then r is said to be what if there is some nonzero element s ∈ R such that rs = 0?
A commutative ring with identity is said to be what if it has no zero divisors?
A commutative ring with identity is said to be what if it has no zero divisors?
What set is a good example of many zero divisors?
What set is a good example of many zero divisors?
What is a Unit?
What is a Unit?
If every nonzero element in a ring R is a unit, then R is called what?
If every nonzero element in a ring R is a unit, then R is called what?
What are Gaussian Integers?
What are Gaussian Integers?
What does the Cancellation Law state?
What does the Cancellation Law state?
What are members of a noncommutative division algebra?
What are members of a noncommutative division algebra?
What is true about every finite integral domain?
What is true about every finite integral domain?
What are the properties of a Field?
What are the properties of a Field?
What are the 3 most notorious Fields?
What are the 3 most notorious Fields?
For any nonnegative integer 'n' and any element 'r' in a ring R, how do we write r + ... + r (n times)?
For any nonnegative integer 'n' and any element 'r' in a ring R, how do we write r + ... + r (n times)?
What is a Characteristic of a ring R?
What is a Characteristic of a ring R?
When can you not use the Cancellation Law?
When can you not use the Cancellation Law?
If 1 has order n in a ring with identity, what is true?
If 1 has order n in a ring with identity, what is true?
What is the characteristic of an integral domain?
What is the characteristic of an integral domain?
What law is satisfied by any dyadic operation ° for which x ° x = x for all elements x in the domain of °?
What law is satisfied by any dyadic operation ° for which x ° x = x for all elements x in the domain of °?
In Boolean algebra, both of the dyadic operations are what?
In Boolean algebra, both of the dyadic operations are what?
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Study Notes
Rings Overview
- A Ring is a nonempty set R with two closed binary operations: addition and multiplication.
- Conditions for a set to be a ring include commutativity in addition, existence of an additive identity (0), existence of additive inverses, and distributive properties of multiplication.
Types of Rings
- A Ring with Multiplicative Identity has an element 1 (1 ≠0) such that 1a = a1 = a for all a ∈ R.
- A Commutative Ring satisfies ab = ba for all a, b ∈ R.
- An Integral Domain is a commutative ring with identity where ab = 0 implies a = 0 or b = 0.
- A Division Ring has an identity where every nonzero element is a unit, allowing for the existence of a multiplicative inverse.
- A Field is a commutative division ring.
Special Elements in Rings
- A Zero Divisor is a nonzero element a in R such that there exists a nonzero element b in R with ab = 0.
- A Unit is an element in R that has a multiplicative inverse.
Substructures and Propositions
- A Subring is a subset S of R that also forms a ring under inherited operations from R.
- Conditions for a subset S to be a subring include S being nonempty, closure under multiplication, and closure under subtraction.
Cancellation and Characteristics
- Proposition 16.10 provides conditions under which a subset is a subring.
- Cancellation Law applies in a commutative ring where for nonzero a, if ab = ac, then b = c; does not hold with zero divisors.
- The Characteristic of a ring is the smallest positive integer n where n(r + r + ... + r) = 0 for all r.
Notable Examples and Theorems
- Gaussian Integers form a ring represented as Z[i] = {m + ni : m, n ∈ Z}.
- Every finite integral domain is classified as a Field.
- Theorem 16.19 states that the characteristic of an integral domain is either prime or zero.
- Quaternions are examples of noncommutative division algebras.
Fields Properties
- Fields have properties such as commutativity in addition and multiplication, the existence of a multiplicative identity, the presence of multiplicative inverses for nonzero elements, and adhere to distributive laws.
- Common fields include the Rationals (Q), Reals (R), and Complex Numbers (C).
Specific Concepts
- Idempotent operations satisfy x ° x = x for all elements x; both operations in Boolean algebra are characterized as idempotent.
- Lemma 16.18 states if 1 has order n in a ring with identity, then the ring's characteristic is n.
Additional Important Terminology
- The notation nr denotes adding an element r in a ring R, n times.
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