Commutative rings with identity

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Definition

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

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)  and  h(1) = 1.

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,·,1), the ring of integers with addition, subtraction, zero, multiplication, and one.

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

[Size 1]?:  1
[Size 2]?:  1
[Size 3]?:  1
[Size 4]?:  4
[Size 5]?:  1
[Size 6]?:  1

Boolean algebras
Integral domains

Superclasses

Commutative rings
Rings with identity

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