Semifield

Remarks

Clauses 3 and 4 say that $S \setminus \set{0}$ is a loop under multiplication. A structure that satisfies clauses 1, 2 and 4 but not necessarily 3 is called a presemifield. If $S$ is finite, clause 4 is equivalent to the absence of zero divisors: $ab = 0$ implies $a = 0$ or $b = 0$. An associative semifield is a division ring, and a finite associative semifield is a finite field by Wedderburn's little theorem.

Example: every field is a semifield, since it satisfies all four clauses and is moreover associative. The real octonions are a semifield in which multiplication is not associative.

Non-example: the ring $\Z$ satisfies clauses 1, 2 and 3 but fails clause 4, because $2x = 1$ has no solution $x \in \Z$.

In another usage, found for instance in tropical geometry, a semifield is a commutative semiring in which every nonzero element has a multiplicative inverse, such as $(\R \cup \set{-\infty}, \max, +)$.