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A non-empty set with two associative binary operations and , satisfying the distributive laws


In most cases one also assumes that the addition is commutative and that there exists a zero such that for every . The most important examples of semi-rings are rings and distributive lattices (cf. Ring; Distributive lattice). If there is a multiplicative identity 1, the two classes are combined by the condition

The non-negative integers with the usual operations provide an example of a semi-ring that does not satisfy this condition.

How to Cite This Entry:
Semi-ring. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by L.A. Skornyakov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article