Near-field

Remarks

Clauses 2 and 3 make multiplication associative on all of $N$, with an identity $1$. Some authors use the left distributive law $a(b + c) = ab + ac$ in clause 4 instead; the two versions are mirror images of each other. A near-field that also satisfies the other distributive law is a division ring. The commutativity of $+$ in clause 1 is often omitted, because it follows from the other clauses.

Example: let $F = \F_9$ be the finite field with $9$ elements, and let $\phi(a) = a^3$ be its Frobenius automorphism. Define $a \circ b = a b$ if $b$ is a square in $F$ (including $b = 0$) and $a \circ b = \phi(a) b$ otherwise. Clause 4 holds because $\phi$ is additive. Clause 2 holds because $\phi$ preserves squares. So $b \circ c$ is a nonzero square exactly when $b$ and $c$ are both squares or both non-squares, and this gives associativity. The element $1$ is the identity. A nonzero $b$ has the inverse $c$ with $\phi^{k}(c) = b^{-1}$, where $k = 0$ if $b$ is a square and $k = 1$ otherwise. Write $\F_9 = \F_3(i)$ with $i^2 = -1$; then $i$ is a square and $1 + i$ is not, and $i \circ (1 + i) = 1 - i$ while $(1 + i) \circ i = i - 1$. So $(F, +, \circ)$ is a near-field with non-commutative multiplication; since every finite division ring is commutative, it is not a division ring.

Non-example: $\Z$ with its usual operations satisfies clauses 1, 3 and 4 but fails clause 2, because $2$ has no multiplicative inverse in $\Z$.