Questions and Answers
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?
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?
Multiplicative Identity
A ring R for which ab = ba for all a, b in R is called what?
Signup and view all the answers
What is an Integral Domain?
Signup and view all the answers
What is a Division Ring?
Signup and view all the answers
What property of a Division Ring is not always required?
Signup and view all the answers
What is a Field?
Signup and view all the answers
What is a Zero Divisor?
Signup and view all the answers
What is a Subring?
Signup and view all the answers
When is a subset S of R considered a subring?
Signup and view all the answers
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?
Signup and view all the answers
A commutative ring with identity is said to be what if it has no zero divisors?
Signup and view all the answers
What set is a good example of many zero divisors?
Signup and view all the answers
What is a Unit?
Signup and view all the answers
If every nonzero element in a ring R is a unit, then R is called what?
Signup and view all the answers
What are Gaussian Integers?
Signup and view all the answers
What does the Cancellation Law state?
Signup and view all the answers
What are members of a noncommutative division algebra?
Signup and view all the answers
What is true about every finite integral domain?
Signup and view all the answers
What are the properties of a Field?
Signup and view all the answers
What are the 3 most notorious Fields?
Signup and view all the answers
For any nonnegative integer 'n' and any element 'r' in a ring R, how do we write r + ... + r (n times)?
Signup and view all the answers
What is a Characteristic of a ring R?
Signup and view all the answers
When can you not use the Cancellation Law?
Signup and view all the answers
If 1 has order n in a ring with identity, what is true?
Signup and view all the answers
What is the characteristic of an integral domain?
Signup and view all the answers
What law is satisfied by any dyadic operation ° for which x ° x = x for all elements x in the domain of °?
Signup and view all the answers
In Boolean algebra, both of the dyadic operations are what?
Signup and view all the answers
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.
Studying That Suits You
Use AI to generate personalized quizzes and flashcards to suit your learning preferences.
Description
Explore the fundamental concepts of rings in mathematics, including the properties and types of rings such as commutative rings, integral domains, and fields. Test your understanding of special elements like zero divisors and units. This quiz covers essential aspects of ring theory.