High Quality Content by WIKIPEDIA articles! High Quality Content by WIKIPEDIA articles! In mathematics, a semifield is an algebraic structure with two binary operations, addition and multiplication, which is similar to the field, but with some axioms relaxed. There are at least two conflicting conventions of what constitutes a semifield. In projective geometry and finite geometry (MSC 51A, 51E, 12K10), a semifield is a ring (S,+,·) where (S,+) is an abelian group with identity element 0, the multiplication is distributive with respect to the addition on the left and on the right, and (S,·) is a division ring that is not assumed to be commutative or associative. This structure is a special case of a quasifield. If S is a finite set, being a division ring is equivalent to the absence of zero divisors, so that a·b = 0 implies that a = 0 or b = 0.

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