Semiring
Algebraic ring that need not have additive negative elements / From Wikipedia, the free encyclopedia
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In abstract algebra, a semiring is an algebraic structure. It is a generalization of a ring, dropping the requirement that each element must have an additive inverse. At the same time, it is a generalization of bounded distributive lattices.
The smallest semiring that is not a ring is the two-element Boolean algebra, e.g. with logical disjunction as addition. A motivating example that is neither a ring nor a lattice is the set of natural numbers under ordinary addition and multiplication, when including the number zero. Semirings are abundant, because a suitable multiplication operation arises as the function composition of endomorphisms over any commutative monoid.
The theory of (associative) algebras over commutative rings can be generalized to one over commutative semirings.[citation needed]