Semiring

A semiring is a set R equipped with two binary operations + and ·, called addition and multiplication, such that:

1. (R, +) is a commutative monoid with identity element 0:
   1. (a + b) + c = a + (b + c)
   2. 0 + a = a + 0 = a
   3. a + b = b + a
2. (R, ·) is a monoid with identity element 1:
   1. (a·b)·c = a·(b·c)
   2. 1·a = a·1 = a
3. Multiplication left and right distributes over addition:
   1. a·(b + c) = (a·b) + (a·c)
   2. (a + b)·c = (a·c) + (b·c)
4. Multiplication by 0 annihilates R:
   1. 0·a = a·0 = 0

This last axiom is omitted from the definition of a ring: it follows automatically from the other ring axioms. Here it does not, and it is necessary to state it in the definition.

The difference between rings and semirings, then, is that addition yields only a commutative monoid, not necessarily a commutative group. Specifically, elements in semirings do not necessarily have an inverse for the addition.

The symbol · is usually omitted from the notation; that is, a·b is just written ab. Similarly, an order of operations is accepted, according to which · is applied before +; that is, a + bc is a + (bc).

A commutative semiring is one whose multiplication is commutative. An idempotent semiring (also known as a dioid) is one whose addition is idempotent: a + a = a, that is, (R, +, 0) is a join-semilattice with zero.

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.