# Generalizing numbers

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.