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near-ring (Definition)

Definitions

A near-ring is a set $ N$ together with two binary operations, denoted $ +\colon N \times N \to N$ and $ \cdot\colon N \times N \to N$, such that

  1. $ (a+b)+c = a+(b+c)$ and $ (a \cdot b) \cdot c = a \cdot (b \cdot c)$ for all $ a,b,c \in N$ (associativity of both operations)
  2. There exists an element $ 0 \in N$ such that $ a+0 = 0+a = a$ for all $ a \in N$ (additive identity)
  3. For all $ a \in N$, there exists $ b \in N$ such that $ a+b = b+a = 0$ (additive inverse)
  4. $ (a+b) \cdot c = (a \cdot c) + (b \cdot c)$ for all $ a,b,c \in N$ (right distributive law)

Note that the axioms of a near-ring differ from those of a ring in that they do not require addition to be commutative, and only require distributivity on one side.

A near-field is a near-ring $ N$ such that $ (N\setminus\{0\},\cdot)$ is a group.

Notes

Every element $ a$ in a near-ring has a unique additive inverse, denoted $ -a$.

We say $ N$ has an identity element if there exists an element $ 1 \in N$ such that $ a \cdot 1 = 1 \cdot a = a$ for all $ a \in N$. We say $ N$ is distributive if $ a\cdot(b+c) = (a \cdot b) + (a \cdot c)$ holds for all $ a,b,c \in N$. We say $ N$ is commutative if $ a \cdot b = b \cdot a$ for all $ a,b \in N$.

Every commutative near-ring is distributive. Every distributive near-ring with an identity element is a unital ring (see the attached proof).

Example

A natural example of a near-ring is the following. Let $ (G,+)$ be a group (not necessarily abelian), and let $ M$ be the set of all functions from $ G$ to $ G$. For two functions $ f$ and $ g$ in $ M$ define $ f+g\in M$ by $ (f+g)(x)=f(x)+g(x)$ for all $ x\in G$. Then $ (M,+,\circ)$ is a near-ring with identity, where $ \circ$ denotes composition of functions.

Bibliography

1
Günter Pilz, Near-Rings, North-Holland, 1983.



"near-ring" is owned by yark. [ full author list (2) | owner history (1) ]
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See Also: ring

Other names:  near ring, nearring
Also defines:  commutative near-ring, commutative near ring, commutative nearring, distributative near-ring, distributative near ring, distributative nearring, near field, nearfield

Attachments:
condition on a near ring to be a ring (Theorem) by CWoo
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Cross-references: composition, functions, unital ring, distributive, identity element, group, distributivity, ring, axioms, right distributive law, operations, associativity, binary operations
There are 5 references to this entry.

This is version 19 of near-ring, born on 2003-02-05, modified 2007-11-04.
Object id is 3968, canonical name is NearRing.
Accessed 6505 times total.

Classification:
AMS MSC: 16Y30 (Associative rings and algebras :: Generalizations :: Near-rings)
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