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 Q {\displaystyle Q} together with two binary operations, + {\displaystyle +} (addition) and ⋅ {\displaystyle \cdot } (multiplication), satisfying the following axioms for all a , b , c {\displaystyle a,b,c} in Q {\displaystyle Q} .
A1: ( Q , + ) {\displaystyle (Q,+)} is an abelian group. A2: ( a ⋅ b ) ⋅ c = a ⋅ ( b ⋅ c ) {\displaystyle (a\cdot b)\cdot c=a\cdot (b\cdot c)} (The associative law for multiplication). A3: ( a + b ) ⋅ c = a ⋅ c + b ⋅ c {\displaystyle (a+b)\cdot c=a\cdot c+b\cdot c} (The right distributive law). A4: Q {\displaystyle Q} contains a non-zero element 1 such that 1 ⋅ a = a ⋅ 1 = a {\displaystyle 1\cdot a=a\cdot 1=a} (Multiplicative identity). A5: For every non-zero element d {\displaystyle d} in Q {\displaystyle Q} there exists an element d − 1 {\displaystyle d^{-1}} such that d ⋅ d − 1 = 1 = d − 1 ⋅ d {\displaystyle d\cdot d^{-1}=1=d^{-1}\cdot d} (Multiplicative inverse).
Notes on the definition The above is, strictly speaking, a definition of a right near-field. By replacing A3 by the left distributive law c ⋅ ( a + b ) = c ⋅ a + c ⋅ b {\displaystyle c\cdot (a+b)=c\cdot a+c\cdot b} 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 x ⋅ 0 = 0 {\displaystyle x\cdot 0=0} for all x {\displaystyle x} ). 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. The exceptional structure can be defined by taking an additive group of order 2, and defining multiplication by x ⋅ y = x {\displaystyle x\cdot y=x} for all x {\displaystyle x} and y {\displaystyle y} .
Examples Any division ring (including any field) is a near-field. The following defines a (right) near-field of order 9. It is the smallest near-field which is not a field. Let K {\displaystyle K} be the Galois field of order 9. Denote multiplication in K {\displaystyle K} by ' ∗ {\displaystyle *} '. Define a new binary operation ' · ' by: If b {\displaystyle b} is any element of K {\displaystyle K} which is a square and a {\displaystyle a} is any element of K {\displaystyle K} then a ⋅ b = a ∗ b {\displaystyle a\cdot b=a*b} . If b {\displaystyle b} is any element of K {\displaystyle K} which is not a square and a {\displaystyle a} is any element of K {\displaystyle K} then a ⋅ b = a 3 ∗ b {\displaystyle a\cdot b=a^{3}*b} . Then K {\displaystyle K} is a near-field with this new multiplication and the same addition as before.
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