Rings Overview and Types

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?

Commutative Ring Signup and view all the answers

What is an Integral Domain?

A commutative ring R with identity where every a, b ∈ R such that ab = 0, and either a = 0 OR b = 0. Signup and view all the answers

What is a Division Ring?

A ring R, with an identity, in which every nonzero element in R is a unit. Signup and view all the answers

What property of a Division Ring is not always required?

Commutativity Signup and view all the answers

What is a Field?

A commutative division ring. Signup and view all the answers

What is a Zero Divisor?

A nonzero element 'a' in a ring R such that there is a nonzero element 'b' in R such that ab = 0. Signup and view all the answers

What is a Subring?

A subset S of R such that S is also a ring under the inherited operations from R. Signup and view all the answers

When is a subset S of R considered a subring?

If and only if S ≠ ∅, rs ∈ S for all r, s ∈ S, and r − s ∈ S for all r, s ∈ S. 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?

Zero Divisor Signup and view all the answers

A commutative ring with identity is said to be what if it has no zero divisors?

Integral Domain Signup and view all the answers

What set is a good example of many zero divisors?

M2(R) Signup and view all the answers

What is a Unit?

An element 'a' in a ring R with identity that has a multiplicative inverse. Signup and view all the answers

If every nonzero element in a ring R is a unit, then R is called what?

Division Ring Signup and view all the answers

What are Gaussian Integers?

A ring such that i^2 = −1 and the set Z[i] = {m + ni : m, n ∈ Z}. Signup and view all the answers

What does the Cancellation Law state?

Let R be a commutative ring with identity. R is an integral domain if and only if for all nonzero elements a ∈ R with ab = ac, we have b=c. Signup and view all the answers

What are members of a noncommutative division algebra?

Quaternions Signup and view all the answers

What is true about every finite integral domain?

Field Signup and view all the answers

What are the properties of a Field?

Addition is commutative, multiplication is commutative, multiplicative identity exists, nonzero elements have multiplicative inverses, and the distributive property holds. Signup and view all the answers

What are the 3 most notorious Fields?

Q: Rationals, R: Reals, C: Complex Numbers 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)?

nr Signup and view all the answers

What is a Characteristic of a ring R?

The least positive integer 'n' such that n(r + r + ... + r) = 0 for all r ∈ R. Signup and view all the answers

When can you not use the Cancellation Law?

Zero Divisors Signup and view all the answers

If 1 has order n in a ring with identity, what is true?

The characteristic of R is n. Signup and view all the answers

What is the characteristic of an integral domain?

Prime or Zero 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 °?

Idempotent Signup and view all the answers

In Boolean algebra, both of the dyadic operations are what?

Idempotent 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.

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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.