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
Δ The abelian group of positive reals under multiplication (isomorphic to the reals under addition via the logarithm) is often denoted \({\mathbb{R}^{+}}\) or \({\mathbb{R}^{\times}}\); both can be potential sources of confusion since the first might seem to imply the operation is addition, and the second does not explicitly signify that the elements must be positive (or alternatively non-zero).

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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