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
MonoidAssociativeUnique identity: \({\textbf{1}a=a\textbf{1}=a}\)
GroupAssociativeInverses: \({aa^{-1}=a^{-1}a=\textbf{1}}\)
RingAbelian groupSemigroupZero: \({\textbf{0}+a=a\Rightarrow a\textbf{0}=\textbf{0}}\)
Integral domainAbelian groupAbelian monoidNo zero divisors
FieldAbelian groupAbelian monoidInverses 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.

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