Near-rings with identity|
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.
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).
0 is a zero for ·: 0·x = 0 and x·0 = 0.
|Congruence n-permutable||yes, n = 2|
|Congruence extension property|
|Definable principal congruences|
|Equationally definable principal congruences|
|Strong amalgamation property|
|Epimorphisms are surjective|