Near-fields

Abbreviation: NFld

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 : M→N that is a homomorphism: h(x + y) = h(x) + h(y)  and  h(x·y) = h(x)·h(y).

Remark: It follows that h(0) = 0 and h(−x) = −h(x).

Some results

0 is a zero for ·: 0·x = 0 and x·0 = 0.

Examples

Properties

Finite members

Subclasses

Superclasses