Unit (ring theory)
The set of units of any ring is closed under multiplication (the product of two units is again a unit), and forms a group for this operation. It never contains the element 0 (except in the case of the trivial ring), and is therefore not closed under addition; its complement however might be a group under addition, which happens if and only if the ring is a local ring.
Unfortunately, the term unit is also used to refer to the identity element 1R of the ring, in expressions like ring with a unit or unit ring, and also e.g. 'unit' matrix. (For this reason, some authors call 1R "unity" or "identity", and say that R is a "ring with unity" or a "ring with identity" rather than a "ring with a unit".)
Group of units
In a commutative unital ring R, the group of units U(R) acts on R via multiplication. The orbits of this action are called sets of associates; in other words, there is an equivalence relation ∼ on R called associatedness such that
- r ∼ s
means that there is a unit u with r = us.
One can check that U is a functor from the category of rings to the category of groups: every ring homomorphism f : R → S induces a group homomorphism U(f) : U(R) → U(S), since f maps units to units. This functor has a left adjoint which is the integral group ring construction.
- In the ring of integers Z, the only units are +1 and −1.
- In the ring Z/nZ of integers modulo n, the units are the congruence classes (mod n) represented by integers coprime to n. They constitute the multiplicative group of integers modulo n.
- Any root of unity in a ring R is a unit. (If rn = 1, then rn − 1 is a multiplicative inverse of r.)
- If R is the ring of integers in a number field, Dirichlet's unit theorem implies that the unit group of R is a finitely generated abelian group. For example, we have (√ + 2)(√ − 2) = 1 in the ring Z[1 + √], and in fact the unit group of this ring is infinite. In general, the unit group of (the ring of integers of) a real quadratic field is infinite (of rank 1).
- The unit group of the ring Mn(F) of n × n matrices over a field F is the group GLn(F) of invertible matrices.
- Dummit, David S.; Foote, Richard M. (2004). Abstract Algebra (3rd ed.). John Wiley & Sons. ISBN 0-471-43334-9.
- Lang, Serge (2002). Algebra. Graduate Texts in Mathematics. Springer. ISBN 0-387-95385-X.
- In a ring, the additive inverse of a non-zero element (1R or else) can equal to the element itself, in which case these statements become a vacuous truth.