Note that the axioms of a near-ring differ from those of a ring in that they do not require addition to be commutative (http://planetmath.org/Commutative), and only require distributivity on one side.
A near-field is a near-ring such that is a group.
Every element in a near-ring has a unique additive inverse, denoted .
We say has an identity element if there exists an element such that for all . We say is distributive if holds for all . We say is commutative if for all .
Every commutative near-ring is distributive. Every distributive near-ring with an identity element is a unital ring (see the attached proof (http://planetmath.org/ConditionOnANearRingToBeARing)).
- 1 Günter Pilz, Near-Rings, North-Holland, 1983.
|Date of creation||2013-03-22 13:25:12|
|Last modified on||2013-03-22 13:25:12|
|Last modified by||yark (2760)|
|Defines||commutative near ring|
|Defines||distributative near ring|