[Home]Commutative rings

HomePage | RecentChanges | Preferences

Abbreviation: CRng

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


HomePage | RecentChanges | Preferences
This page is read-only | View other revisions
Last edited November 29, 2003 4:43 pm (diff)
Search: