Math 2.0

Field theory and polynomials Definition

Division ring

Builds on RingUnit of a ring

Definition

A division ring is a ring DD with these two properties.

  1. 1≠01 \neq 0 in DD.
  2. Every nonzero element of DD is a unit: for each a≠0a \neq 0 in DD, there is b∈Db \in D with ab=ba=1a b = b a = 1.

Remarks

A commutative division ring is a field. The quaternions are a division ring that is not commutative. By Wedderburn's little theorem, every finite division ring is a field.

Reading path

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Group, concept, no evidence levelGroupconceptRing, definition, no evidence levelRingdefinitionUnit of a ring, definition, no evidence levelUnit of a ringdefinitionDivision ring, definition, no evidence levelDivision ringdefinition
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  1. Group concept
  2. Ring definition
  3. Unit of a ring definition
  4. Division ring definition

Where this leads

36 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Division_ring, lead section
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DefinesDivision ring
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