# nLab semiring

Contents

### Context

#### Algebra

higher algebra

universal algebra

# Contents

## Definitions

Similarily to the definition of a ring, mathematicians disagree on the definition of a semiring. There are 4 different definitions of a semiring used in mathematics:

1. A semigroup object in the category of commutative semigroups
2. A monoid object in the category of commutative semigroups
3. A semigroup object in the category of commutative monoids
4. A monoid object in the category of commutative monoids

The last algebraic structure is also called a rig.

Historically, rings were defined without a multiplicative identity element, and the primary definition of semiring used was the third definition.

However, more recently, rings have been defined with multiplicative units, and thus nowadays the primary definition of semiring used is the fourth definition. In this context, semirings and rings lacking multiplicative identity elements are called nonunital.

Moreover, some mathematicians argue that semiring should be used in analogy to semigroup, where semirings do not have additive identity elements, and semirings with 0 are instead called rigs, resulting in the first and second definitions of a semiring respectively.

The first definition is the most general of the four definitions of a semiring, and some authors use it precisely because of its generality.

If one adopts the second definition to define a semiring, and the fourth definition to define a rig, then the first and third are then called nonunital semirings and nonunital rigs respectively.

## References

Last revised on August 18, 2022 at 18:01:26. See the history of this page for a list of all contributions to it.