# Near-fields

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 Near-rings
 Near-rings with identity

### Definition

A near-field is a near-ring with identity N = (N, + ,−,0,·,1 ) such that
N is non-trivial:   0 ≠ 1 and
every non-zero element has a multiplicative inverse:   x ≠ 0  ⇒  ∃y (x·y = 1).

Remark: The inverse of x is unique, and is usually denoted by x−1.

### Morphisms

Let M and N be near-fields. A morphism from M to N is a function h : MN 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 = 0 and x·0 = 0.

### 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

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Fields

### Superclasses

Near-rings with identity

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