Nearring
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In mathematics, a near-ring (also near ring or nearring) is an algebraic structure similar to a ring but satisfying fewer properties. Near-rings arise naturally from functions on groups.
[edit] Abstract definition
A set N together with two binary operations + (called addition) and ⋅ (called multiplication) is called a (right) near-ring if:
- N is a group (not necessarily abelian!) under addition;
- multiplication is associative (so N is a semigroup under multiplication); and
- multiplication distributes over addition on the right: for any x, y, z in N, it holds that (x + y) ⋅ z = (x ⋅ z) + (y ⋅ z).
An immediate consequence of this one-sided distributive law is that it is true that 0 ⋅ x = 0 but it may also be true that x ⋅ 0 ≠ 0 for an x in N. A near-ring is a ring if and only if addition is commutative and multiplication is distributive over addition on the left.
[edit] The near-ring of mappings of a group
Let G be a group (or even monoid), written additively but may be nonabelian, and let M(G) be the set {f | f : G → G} of all functions from G to G. An addition operation can be defined on M(G): given f, g in M(G), then the mapping f + g from G to G is given by (f + g)(x) = f(x) + g(x) for all x in G. Then M(G) is an additive group, which is abelian if and only if G is. Taking the composition of mappings as the product ⋅, M(G) becomes a prototypical near-ring. One then notices that the axioms above are satisfied.
The 0 element of the near-ring M(G) is the zero map, i.e., the mapping which takes every element of G to the zero element of G.
If G has at least 2 elements, M(G) is never a ring.
