Commutative regular rings

Abbreviation: CRRng

Definition

A commutative regular ring is a regular ring R = (R, + ,−,0,·,1 ) such that · is commutative: x·y = y·x.

Morphisms

Let R and S be commutative regular rings. 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) and h(1) = 1.

Some results

Examples

Properties

Classtype	first-order
Equational theory
Quasiequational theory
First-order theory
Locally finite	no
Residual size	unbounded
Congruence distributive
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

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Subclasses

Fields

Superclasses

Commutative rings with identity