Near-Field (Mathematics)

# Near-field (mathematics)

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Description:
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

A near-field is a set [itex]Q[/itex], together with two binary operations, ' + ' (addition) and ' • ' (multiplication), satisfying the following axioms:
A1: [itex](Q, +)[/itex] is an Abelian group
A2: [itex](a[/itex] • [itex]b)[/itex] • [itex]c[/itex] = [itex]a[/itex] • [itex](b[/itex] • [itex]c)[/itex] for all elements [itex]a[/itex], [itex]b[/itex], [itex]c[/itex] of [itex]Q[/itex] (The associative law for multiplication)
A3: [itex](a + b) [/itex] • [itex] c = a [/itex] • [itex]c + b [/itex] • [itex] c[/itex] for all elements [itex]a[/itex], [itex]b[/itex], [itex]c[/itex] of [itex]Q[/itex] (The right distributive law)
A4: [itex]Q[/itex] contains an element 1 such that [itex]1 [/itex] • [itex] a = a [/itex] • [itex] 1 = a[/itex] for every element [itex]a[/itex] of [itex]Q[/itex] (Multiplicative identity)
A5: For every non-zero element a of......
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