Near-Field (Mathematics)

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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 <math>Q</math>, together with two binary operations, ' + ' (addition) and ' • ' (multiplication), satisfying the following axioms:

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- A1: <math>(Q, +)</math> is an Abelian group
- A2: <math>(a</math> • <math>b)</math> • <math>c</math> = <math>a</math> • <math>(b</math> • <math>c)</math> for all elements <math>a</math>, <math>b</math>, <math>c</math> of <math>Q</math> (The associative law for multiplication)
- A3: <math>(a + b) </math> • <math> c = a </math> • <math>c + b </math> • <math> c</math> for all elements <math>a</math>, <math>b</math>, <math>c</math> of <math>Q</math> (The right distributive law)
- A4: <math>Q</math> contains an element 1 such that <math>1 </math> • <math> a = a </math> • <math> 1 = a</math> for every element <math>a</math> of <math>Q</math> (Multiplicative identity)
- A5: For every non-zero element a of...... ...

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