# Commutative rings

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Difference (from revision 11 to current revision) (minor diff, author diff)

Changed: 8,10c8,10
 Note: Idl(R) = { all ideals of R}, I is an ideal if a,b ∈ I  ⇒  a + b ∈ I and ∀r  ∈ R { r·I ⊆ I, I·r ⊆ I
 Remark: Idl(R) = { all ideals of R} I is an ideal if a,b ∈ I  ⇒  a + b ∈ I and ∀r  ∈ R (r·I ⊆ I)

### Definition

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

Remark: Idl(R) = { all ideals of 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 : RS that is a homomorphism: h(x + y) = h(x) + h(y)  and  h(x·y) = h(xh(y).

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

### Some results

0 is a zero for ·: x = x and x·0 = 0.

### Examples

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

### Properties

 Classtype variety Equational theory decidable Quasiequational theory First-order theory undecidable Locally finite no Residual size unbounded Congruence distributive no Congruence modular yes Congruence n-permutable yes, n = 2 Congruence regular yes Congruence uniform yes Congruence extension property Definable principal congruences Equationally definable principal congruences Amalgamation property Strong amalgamation property Epimorphisms are surjective

### 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

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