[Home]Rings with identity

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Abbreviation: Rng1

Definition

A ring with identity is a structure R = (R, + ,−,0,·,1 ) of type (2,1,0,2,0) such that
(R, + ,−,0,·) is a ring and
1 is an identity for ·:   x·1 = x  and  1·x = x.

Morphisms

Let R and S be 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 = 0 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

Finite members

[Size 1]?:  1
[Size 2]?:  1
[Size 3]?:  1
[Size 4]?:  4
[Size 5]?:  1
[Size 6]?:  1
[Finite rings with identity in the Encyclopedia of Integer Sequences]

Subclasses

Commutative rings with identity
Division rings

Superclasses

Rings


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Last edited April 19, 2003 10:47 am (diff)
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