Nearring
From Wikipedia, the free encyclopedia
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.
Contents |
[edit] Definition
A set N together with two binary operations + (called addition) and ⋅ (called multiplication) is called a (right) near-ring if:
- A1: N is a group (not necessarily abelian) under addition;
- A2: multiplication is associative (so N is a semigroup under multiplication); and
- A3: multiplication distributes over addition on the right: for any x, y, z in N, it holds that (x + y) ⋅ z = (x ⋅ z) + (y ⋅ z).[1]
Similarly it is possible to define a left near-ring by replacing the right distributive law A3 by the corresponding left distributive law. However, near-rings are almost always written as right near-rings.
An immediate consequence of this one-sided distributive law is that it is true that 0 ⋅ x = 0 but it is not necessarily true that x ⋅ 0 = 0 for any x in N. Another immediate consequence is that (- x) ⋅ y = - (x ⋅ y) for any x, y 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] Mappings from a group to itself
For an additive, but possibly nonabelian, group G, 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 abelian. Taking the composition of mappings as the product ⋅, M(G) becomes a near-ring.
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 identity element of G.
As with addition and 0, the additive inverse −f of f in M(G) is also defined pointwise, that is (−f)(x) = −(f(x)) for all x in G.
If G has at least 2 elements, M(G) is not a ring.
Many subsets of M(G) form interesting and useful near-rings. For example:[2]
- The mappings for which f(0) = 0
- The constant mappings, i.e. those which map every element of the group to one fixed element
- The set of maps generated by addition and negation from the endomorphisms of the group (the "additive closure" of the set of endomorphisms).
Further examples are to be found if the group has further structure, for example:
- The continuous mappings in a topological group
- The polynomial functions on a ring with identity
- The affine maps in a vector space.
Every nearring is isomorphic to a sub-nearring of M(G) for some G.
[edit] Applications
Many applications involve the subclass of nearrings known as near fields; for these see the article on near fields.
There are various applications of proper near-rings, i.e. those which are neither rings nor near-fields. The best known is to balanced incomplete block designs[3]
[edit] See also
[edit] References
- ^ [M. Farag, (2007), "Near-Rings: What They Are and What They Are Good For" in Proceedings of the 21st Summer Research Institute, Australian Mathematical Society AMS Bookstore (1981), ISBN 0821850091, 9780821850091 ]
- ^ [M. Farag, (2007), "Near-Rings: What They Are and What They Are Good For" in Proceedings of the 21st Summer Research Institute, Australian Mathematical Society AMS Bookstore (1981), ISBN 0821850091, 9780821850091]
- ^ [G. Pilz " Near-rings", North-Holland, Amsterdam, second edition, (1983)]
