Definition 2.7.1.  Let R be a set, ⊕ : R × R → R be an operation on R, 0 ∈ R, ⊙ : R × R → R be an operation on R, 1 ∈ R. We define:
(R, ⊕, 0, ⊙, 1) forms a semiring  :⇔  (R, ⊕, 0) forms a commutative monoid and (R, ⊙, 1) forms a monoid and ⊙ is distributive over ⊕ and 0 is an absorbing element for ⊙
References.