Nearring
Encyclopedia
In mathematics
, a near-ring (also near ring or nearring) is an algebraic
structure
similar to a ring, but that satisfies fewer axioms. Near-rings arise naturally from functions
on group
s.
s + (called addition) and ⋅ (called multiplication) is called a (right) near-ring if:
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, but it is not necessary that x ⋅ (- y) = - (x ⋅ y). A near-ring is a ring
(not necessarily with unity) if and only if
addition is commutative and multiplication is distributive over addition on the left.
, 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 also a 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 that takes every element of G to the identity element of G. The additive inverse −f of f in M(G) coincides with the natural pointwise definition, that is, (−f)(x) = −(f(x)) for all x in G.
If G has at least 2 elements, M(G) is not a ring, even if G is abelian. (Consider the constant mapping g from G to a fixed element g≠0 of G; g·0 = g ≠ 0.) However, there is a subset E(G) of M(G) consisting of all group endomorphism
s of G, that is, all maps f : G → G such that f(x + y) = f(x) + f(y) for all x, y in G. If (G, +) is abelian, both near-ring operations on M(G) are closed on E(G), and (E(G), +, ⋅) is a ring. If (G, +) is nonabelian, E(G) is generally not closed under the near-ring operations; but any subset of M(G) that contains E(G) and is closed under the near-ring operations is also a near-ring.
Many subsets of M(G) form interesting and useful near-rings. For example:
Further examples occur if the group has further structure, for example:
Every nearring is isomorphic
to a sub-nearring of M(G) for some G.
; for these see the article on near fields.
There are various applications for proper near-rings, i.e., those that are neither rings nor near-fields.
The best known is to balanced incomplete block designs using planar nearrings.
These are a way to obtain Difference Families using the orbits of a fixed point free automorphism group of a group. Clay and others have extended these ideas to more general geometrical constructions
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a near-ring (also near ring or nearring) is an algebraic
Abstract algebra
Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras...
structure
Algebraic structure
In abstract algebra, an algebraic structure consists of one or more sets, called underlying sets or carriers or sorts, closed under one or more operations, satisfying some axioms. Abstract algebra is primarily the study of algebraic structures and their properties...
similar to a ring, but that satisfies fewer axioms. Near-rings arise naturally from functions
Function (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
on group
Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
s.
Definition
A set N together with two binary operationBinary operation
In mathematics, a binary operation is a calculation involving two operands, in other words, an operation whose arity is two. Examples include the familiar arithmetic operations of addition, subtraction, multiplication and division....
s + (called addition) and ⋅ (called multiplication) is called a (right) near-ring if:
- A1: N is a group (not necessarily abelianAbelian groupIn abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order . Abelian groups generalize the arithmetic of addition of integers...
) under addition; - A2: multiplication is associative (so N is a semigroupSemigroupIn mathematics, a semigroup is an algebraic structure consisting of a set together with an associative binary operation. A semigroup generalizes a monoid in that there might not exist an identity element...
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).G. Pilz, (1982), "Near-Rings: What They Are and What They Are Good For" in Contemp. Math., 9, pp. 97--119. Amer. Math. Soc., Providence, R.I., 1981.
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, but it is not necessary that x ⋅ (- y) = - (x ⋅ y). A near-ring is a ring
Ring theory
In abstract algebra, ring theory is the study of rings—algebraic structures in which addition and multiplication are defined and have similar properties to those familiar from the integers...
(not necessarily with unity) if and only if
If and only if
In logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements....
addition is commutative and multiplication is distributive over addition on the left.
Mappings from a group to itself
Let G be a group, written additively but not necessarily abelianAbelian group
In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order . Abelian groups generalize the arithmetic of addition of integers...
, 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 also a 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 that takes every element of G to the identity element of G. The additive inverse −f of f in M(G) coincides with the natural pointwise definition, that is, (−f)(x) = −(f(x)) for all x in G.
If G has at least 2 elements, M(G) is not a ring, even if G is abelian. (Consider the constant mapping g from G to a fixed element g≠0 of G; g·0 = g ≠ 0.) However, there is a subset E(G) of M(G) consisting of all group endomorphism
Endomorphism
In mathematics, an endomorphism is a morphism from a mathematical object to itself. For example, an endomorphism of a vector space V is a linear map ƒ: V → V, and an endomorphism of a group G is a group homomorphism ƒ: G → G. In general, we can talk about...
s of G, that is, all maps f : G → G such that f(x + y) = f(x) + f(y) for all x, y in G. If (G, +) is abelian, both near-ring operations on M(G) are closed on E(G), and (E(G), +, ⋅) is a ring. If (G, +) is nonabelian, E(G) is generally not closed under the near-ring operations; but any subset of M(G) that contains E(G) and is closed under the near-ring operations is also a near-ring.
Many subsets of M(G) form interesting and useful near-rings. For example:
- The mappings for which f(0) = 0
- The constant mappings, i.e., those that map every element of the group to one fixed element
- The set of maps generated by addition and negation from the endomorphismsEndomorphismIn mathematics, an endomorphism is a morphism from a mathematical object to itself. For example, an endomorphism of a vector space V is a linear map ƒ: V → V, and an endomorphism of a group G is a group homomorphism ƒ: G → G. In general, we can talk about...
of the group (the "additive closure" of the set of endomorphisms).
Further examples occur if the group has further structure, for example:
- The continuous mappings in a topological groupTopological groupIn mathematics, a topological group is a group G together with a topology on G such that the group's binary operation and the group's inverse function are continuous functions with respect to the topology. A topological group is a mathematical object with both an algebraic structure and a...
- The polynomial functions on a ring with identity under addition and polynomial composition
- The affine maps in a vector spaceVector spaceA vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
.
Every nearring is isomorphic
Isomorphism
In abstract algebra, an isomorphism is a mapping between objects that shows a relationship between two properties or operations. If there exists an isomorphism between two structures, the two structures are said to be isomorphic. In a certain sense, isomorphic structures are...
to a sub-nearring of M(G) for some G.
Applications
Many applications involve the subclass of nearrings known as near fieldsNear-field (mathematics)
In mathematics, a near-field is an algebraic structure similar to a division ring, except that it has only one of the two distributive laws. Alternatively, a near-field is a near-ring in which there is a multiplicative identity, and every non-zero element has a multiplicative inverse.- Definition...
; for these see the article on near fields.
There are various applications for proper near-rings, i.e., those that are neither rings nor near-fields.
The best known is to balanced incomplete block designs using planar nearrings.
These are a way to obtain Difference Families using the orbits of a fixed point free automorphism group of a group. Clay and others have extended these ideas to more general geometrical constructions
External links
- The Near Ring Main Page at the Johannes Kepler Universität Linz