lacunary - Mathnotes

Rings

Definition: Ring \@{ring}

A ring is a set together with two binary operations + and ⋅, which we will call addition and multiplication, such that the following axioms are satisfied:

  1. ⟨R,+⟩ is an Abelian group.
  1. Multiplication is associative.
  1. For all a,b,c∈R, the left distributive law and the right distributive law hold, i.e.

a⋅(b+c)=(a⋅b)+(a⋅c),(a+b)⋅c=(a⋅c)+(b⋅c).

Referenced by (3 direct, 371 transitive)

Direct references:

Transitive (depth 1):

Transitive (depth 2):

Transitive (depth 3):

Transitive (depth 4):

Transitive (depth 5):

Transitive (depth 6):

Transitive (depth 7):

Transitive (depth 8):

Transitive (depth 9):

Transitive (depth 10):

Transitive (depth 11):

Example: Examples of Rings \@{example-2}

The integers, rationals, reals and complex numbers are all rings with the usual addition and multiplication.

Definition: Ring Homomorphism \@{ring-homomorphism}

A ring homomorphism ϕ:R→R′ must satisfy the following two properties:

  1. ϕ(a+b)=ϕ(a)+ϕ(b).
  1. ϕ(ab)=ϕ(a)ϕ(b).
Remark \@{ring-multiplicative-identity}

A ring doesn't have to have a multiplicative identity element, but it can. If it has one, it's denoted 1 and for all a in R satisfies a1=1a=a.

Definition: unity \@{unity}

The element 1 is also called unity.

Referenced by (4 direct, 370 transitive)

Direct references:

Transitive (depth 1):

Transitive (depth 2):

Transitive (depth 3):

Transitive (depth 4):

Transitive (depth 5):

Transitive (depth 6):

Transitive (depth 7):

Transitive (depth 8):

Transitive (depth 9):

Transitive (depth 10):

Transitive (depth 11):

Definition: Commutative Ring \@{commutative-ring}

A ring in which multiplication is commutative is called a commutative ring.

Definition: Ring with Unity \@{ring-with-unity}

A ring that has a multiplicative identity element is called a ring with unity.

Referenced by (2 direct, 369 transitive)

Direct references:

Transitive (depth 1):

Transitive (depth 2):

Transitive (depth 3):

Transitive (depth 4):

Transitive (depth 5):

Transitive (depth 6):

Transitive (depth 7):

Transitive (depth 8):

Transitive (depth 9):

Transitive (depth 10):

Definition: Multiplicative Inverse \@{multiplicative-inverse}

For some element a in a ring with unity R where 1≠0, if a−1∈R such that aa−1=a−1a=1, a−1 is said to be the multiplicative inverse of a.

Referenced by (2 direct, 368 transitive)

Direct references:

Transitive (depth 1):

Transitive (depth 2):

Transitive (depth 3):

Transitive (depth 4):

Transitive (depth 5):

Transitive (depth 6):

Transitive (depth 7):

Transitive (depth 8):

Transitive (depth 9):

Transitive (depth 10):

Definition: Unit \@{unit}

If a has a multiplicative inverse in R, a is said to be a unit in R.

Referenced by (3 direct, 366 transitive)

Direct references:

Transitive (depth 1):

Transitive (depth 2):

Transitive (depth 3):

Transitive (depth 4):

Transitive (depth 5):

Transitive (depth 6):

Transitive (depth 7):

Transitive (depth 8):

Transitive (depth 9):

Transitive (depth 10):

Fields

Definition: Division Ring \@{division-ring}

Let R be a ring with unity. If every nonzero element of R is a unit (has a multiplicative inverse), then R is called a division ring.

Referenced by (1 direct, 366 transitive)

Direct references:

Transitive (depth 1):

Transitive (depth 2):

Transitive (depth 3):

Transitive (depth 4):

Transitive (depth 5):

Transitive (depth 6):

Transitive (depth 7):

Transitive (depth 8):

Transitive (depth 9):

Definition: Field \@{field}

A commutative division ring is called a field.

Referenced by (5 direct, 361 transitive)

Direct references:

Transitive (depth 1):

Transitive (depth 2):

Transitive (depth 3):

Transitive (depth 4):

Transitive (depth 5):

Transitive (depth 6):

Transitive (depth 7):

Transitive (depth 8):

Example: Examples of Fields \@{example-12}

The integers are not a field, but the rationals, the reals, and the complex numbers are all fields.