Rings and FieldsMain articles: ring (mathematics) and field (mathematics) See also: Ring theory, Glossary of ring theory, Field theory (mathematics), and glossary of field theory
Groups just have one binary operation. To fully explain the behaviour of the different types of numbers, structures with two operators need to be studied. The most important of these are rings, and fields.
A ring has two binary operations (+) and (×), with × distributive over +. Under the first operator (+) it forms an abelian group. Under the second operator (×) it is associative, but it does not need to have identity, or inverse, so division is not required. The additive (+) identity element is written as 0 and the additive inverse of a is written as −a.
Distributivity generalises the distributive law for numbers, and specifies the order in which the operators should be applied, (called the precedence). For the integers (a + b) × c = a × c + b × c and c × (a + b) = c × a + c × b, and × is said to be distributive over +.
The integers are an example of a ring. The integers have additional properties which make it an integral domain.
A field is a ring with the additional property that all the elements excluding 0 form an abelian group under ×. The multiplicative (×) identity is written as 1 and the multiplicative inverse of a is written as a−1.
The rational numbers, the real numbers and the complex numbers are all examples of fields.
Other articles related to "rings and fields, rings, ring, and field, field, and fields":
... The signature of (unital) rings has 2 constants 0 and 1, two binary functions + and ×, and, optionally, one unary inverse functions − −1 ... Rings Axioms Addition makes the ring into an abelian group, multiplication is associative and has an identity 1, and multiplication is left and right distributive ... Commutative rings The axioms for rings plus ∀x ∀y xy=yx ...
... Main articles ring (mathematics) and field (mathematics) See also Ring theory, Glossary of ring theory, Field theory (mathematics), and glossary of field theory Groups just have one binary operation ... The most important of these are rings, and fields ... A ring has two binary operations (+) and (×), with × distributive over + ...
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