# Semi-ring

From Encyclopedia of Mathematics

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

and

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: http://encyclopediaofmath.org/index.php?title=Semi-ring&oldid=16685

This article was adapted from an original article by L.A. Skornyakov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article