Math 2.0

Associative rings and algebras Definition

Group of units of a ring

Builds on Unit of a ringGroup

Definition

Let RR be a ring. The group of units of RR is the set R×R^\times of the units of RR, with the multiplication of RR. It is a group with identity element 11.

Remarks

The set R×R^\times is closed under multiplication: if uu and vv are units, then (uv)−1=v−1u−1(u v)^{-1} = v^{-1} u^{-1}. Other notations for R×R^\times are R∗R^* and U(R)U(R). The group ℤ×={1,−1}\mathbb{Z}^\times = \{1, -1\} has order 22. For a field KK, K×=K∖{0}K^\times = K \setminus \{0\}.

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17 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Unit_(ring_theory), Group of units
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