quasiring

A quasiring is an algebraic structure that generalizes the notion of a ring by relaxing some of the ring axioms, particularly the requirement that every element has an additive inverse. The term is used in different ways in the literature, and its exact definition can vary. In many contexts a quasiring is essentially a semiring with two binary operations that behave like addition and multiplication, but without the necessity of additive inverses.

Formally, a quasiring consists of a nonempty set Q equipped with two binary operations, addition and multiplication, satisfying several standard ring-like laws. These typically include that (Q, +) is a commutative monoid with identity element 0, and (Q, ×) is a monoid with identity element 1 (though some authors omit requiring a multiplicative identity). Multiplication distributes over addition on both sides: a×(b+c) = a×b + a×c and (b+c)×a = b×a + c×a for all a, b, c in Q. It is also common to require that 0 acts as a multiplicative annihilator, so 0×a = a×0 = 0 for all a in Q. Additive inverses need not exist, which is what distinguishes a quasiring from a ring.

Variants and relationships: If a quasiring includes a multiplicative identity and additive inverses for every element, it is a ring. If it lacks additive inverses but retains the other laws, it is often called a semiring or rig in contemporary usage; some authors still call such a structure a quasiring. The terminology is not universal, and some texts prefer “semiring” over “quasiring.”

Examples: The natural numbers with usual addition and multiplication form a semiring (and thus a quasiring under several definitions). Boolean algebras with OR as addition and AND as multiplication also serve as examples of semiring-like structures.

See also: Semiring, Ring, rng, monoid, distributive law.