Concept
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. A near-field is a set together with two binary operations, (addition) and (multiplication), satisfying the following axioms: A1: is an abelian group. A2: = for all elements , , of (The associative law for multiplication). A3: for all elements , , of (The right distributive law). A4: contains an element 1 such that for every element of (Multiplicative identity). A5: For every non-zero element of there exists an element such that (Multiplicative inverse). The above is, strictly speaking, a definition of a right near-field. By replacing A3 by the left distributive law we get a left near-field instead. Most commonly, "near-field" is taken as meaning "right near-field", but this is not a universal convention. A (right) near-field is called "planar" if it is also a right quasifield. Every finite near-field is planar, but infinite near-fields need not be. It is not necessary to specify that the additive group is abelian, as this follows from the other axioms, as proved by B.H. Neumann and J.L. Zemmer. However, the proof is quite difficult, and it is more convenient to include this in the axioms so that progress with establishing the properties of near-fields can start more rapidly. Sometimes a list of axioms is given in which A4 and A5 are replaced by the following single statement: A4*: The non-zero elements form a group under multiplication. However, this alternative definition includes one exceptional structure of order 2 which fails to satisfy various basic theorems (such as for all ). Thus it is much more convenient, and more usual, to use the axioms in the form given above. The difference is that A4 requires 1 to be an identity for all elements, A4* only for non-zero elements.