Algebra is concerned with sets and operations on these sets. The most common algebraic objects can be viewed as generalizations of the two most familiar examples: the integers and real numbers under addition and multiplication. The generalization starts with a plain set and incrementally adds the properties that define \({\mathbb{R}}\), yielding objects with increasing structure.

Addition \({+}\) | Multiplication \({\times}\) | Special features | |
---|---|---|---|

Semigroup | Associative | ||

Monoid | Associative | Unique identity: \({\textbf{1}a=a\textbf{1}=a}\) | |

Group | Associative | Inverses: \({aa^{-1}=a^{-1}a=\textbf{1}}\) | |

Ring | Abelian group | Semigroup | Zero: \({\textbf{0}+a=a\Rightarrow a\textbf{0}=\textbf{0}}\) |

Integral domain | Abelian group | Abelian monoid | No zero divisors |

Field | Abelian group | Abelian monoid | Inverses under \({\times}\) except for \({\textbf{0}}\) |

Notes: \({a\times b}\) is denoted \({ab}\) and the identity under \({\times}\) is \({\mathbf{1}}\) (other common notations include \({I}\) and \({e}\)). For a ring the identity under \({+}\) is denoted \({\mathbf{0}}\) and called zero. Abelian, AKA commutative, means \({ab=ba}\). The ring operation \({\times}\) is distributive over \({+}\). No zero divisors means \({ab=\mathbf{0}}\) only if \({a=\mathbf{0}}\) or \({b=\mathbf{0}}\).

Δ It is important to distinguish abstract operations and elements from “ordinary” ones in a particular case. For example, the integers are a group over ordinary addition, so that \({a+b}\) could be written \({ab}\) and 0 denoted \({\mathbf{1}}\) in a group context. On the other hand, abelian groups are usually written using \({+}\) as the operation instead of \({\times}\), with “integer multiplication” defined as \({na\equiv a+a+\dotsb+a}\) (\({n}\) times); integer multiplication should not be confused with the multiplication of a ring structure, which may be different. |

Immediate examples are the real numbers as a field, and the integers as an integral domain; however, the integers are not a group under multiplication since only 1 has an inverse. Some further examples can help illuminate the boundaries between these structures.

- Semigroup but not monoid: the positive reals less than 1 under multiplication (no identity)
- Monoid but not group: the integers under multiplication (no inverses)
- Group but not abelian group: real matrices with non-zero determinant under multiplication
- Abelian group: the integers or real numbers under addition
- Ring but not integral domain: the ring of integers mod \({n}\) for \({n}\) not prime (zero divisor \({pq=n=0}\))
- Integral domain but not field: the integers (no multiplicative inverses)
- Field: the real numbers; the complex numbers

A **generating set** of an algebraic object is a subset of elements that lead to any other element via operations (e.g. \({+}\), \({{\times}}\)). The subset **generates** the object, and the elements in the subset are **generators**. An abelian group is called **finitely generated** if it has a finite generating set.