Examples
- In the ring of integers Z, the units are +1 and −1. The associates are pairs n and −n.
- In the ring of integers modulo n, Z/nZ, the units are the congruence classes (mod n) which are coprime to n. They constitute the multiplicative group of integers (mod n).
- Any root of unity is a unit in any unital ring R. (If r is a root of unity, and rn = 1, then r−1 = rn − 1 is also an element of R by closure under multiplication.)
- If R is the ring of integers in a number field, Dirichlet's unit theorem states that the group of units of R is a finitely generated abelian group. For example, we have (√5 + 2)(√5 − 2) = 1 in the ring of integers of Q, and in fact the unit group is infinite in this case. In general, the unit group of a real quadratic field is always infinite (of rank 1).
- In the ring M(n,F) of n×n matrices over a field F, the units are exactly the invertible matrices.
Read more about this topic: Unit (ring Theory)
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