HomePage | RecentChanges | Preferences

#### Abbreviation: CRng

### Definition

A *commutative ring* is a ring *R* = (*R*, + ,−,0,·) such that
· is commutative: *x*·*y* = *y* ·*x*.

**Remark**: *I**d**l*(*R*) = { *a**l**l* *i**d**e**a**l**s* *o**f* *R*}

*I* is an ideal if *a*,*b* ∈ *I* ⇒ *a* + *b* ∈ *I*

and ∀*r* ∈ *R* (*r*·*I* ⊆ *I*)

### Morphisms

Let *R* and *S* be commutative rings with identity. A morphism from *R*
to *S* is a function *h* : *R*→*S* that is a homomorphism:
*h*(*x* + *y*) = *h*(*x*) + *h*(*y*) and *h*(*x*·*y*) = *h*(*x*)·*h*(*y*).

**Remark**:
It follows that *h*(0) = 0 and *h*(−*x*) = −*h*(*x*).

### Some results

0 is a zero for ·: 0·*x* = *x* and *x*·0 = 0.

### Examples

(**Z**, + ,−,0,·), the ring of integers with addition, subtraction, zero, and multiplication.

### Properties

### Finite members

[Size 1]?: 1

[Size 2]?: 2

[Size 3]?: 2

[Size 4]?: 9

[Size 5]?: 2

[Size 6]?: 4

[Finite commutative rings in the Encyclopedia of Integer Sequences]

### Subclasses

Commutative rings with identity

Fields

### Superclasses

Rings