Compared to a ring, a semiring omits the requirement for inverses under addition; that is, it requires only a commutative monoid, not a commutative group. In a ring, the additive inverse requirement implies the existence of a multiplicative zero, so here it must be specified explicitly. If a semiring's multiplication is commutative, then it is called a commutative semiring.
There are some authors who prefer to leave out the requirement that a semiring have a 0 or 1. This makes the analogy between ring and semiring on the one hand and group and semigroup on the other hand work more smoothly. These authors often use rig for the concept defined here.[note 1]
Examples of a complete semiring are the power set of a monoid under union and the matrix semiring over a complete semiring.
The term dioid (for "double monoid") has been used to mean various types of semirings:
A generalization of semirings does not require the existence of a multiplicative identity, so that multiplication is a semigroup rather than a monoid. Such structures are called hemirings or pre-semirings. A further generalization are left-pre-semirings, which additionally do not require right-distributivity (or right-pre-semirings, which do not require left-distributivity).
Yet a further generalization are near-semirings: in addition to not requiring a neutral element for product, or right-distributivity (or left-distributivity), they do not require addition to be commutative. Just as cardinal numbers form a (class) semiring, so do ordinal numbers form a near-semiring, when the standard ordinal addition and multiplication are taken into account. However, the class of ordinals can be turned into a semiring by considering the so-called natural (or Hessenberg) operations instead.
In category theory, a 2-rig is a category with functorial operations analogous to those of a rig. That the cardinal numbers form a rig can be categorified to say that the category of sets (or more generally, any topos) is a 2-rig.