Unit (ring theory)

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Not to be confused with Unit ring.

In mathematics, an invertible element or a unit in a (unital) ring R is any element u that has an inverse element in the multiplicative monoid of R, i.e. an element v such that

uv = vu = 1R, where 1R is the multiplicative identity.[1][2]

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 zero 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.

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".

The multiplicative identity 1R and its opposite −1R are always units. Hence, pairs of additive inverse elements[3] x and x are always associated.

Examples[edit]

In any ring, 1 is a unit. More generally, any root of unity in a ring R is a unit: if rn = 1, then rn − 1 is a multiplicative inverse of r. On the other hand, 0 is never a unit. A ring R is a field (possibly non-commutative, also known as a skew field or division ring) if and only if U(R) = R ∖ {0}. For example, the units of the real numbers R are R ∖ {0}. Thus, for any ring R, there is an inclusion

Integers[edit]

In the ring of integers Z, the only units are +1 and −1.

Rings of integers in a number field F have, in general, more units. For example,

(5 + 2)(5 − 2) = 1

in the ring Z[1 + 5/2], and in fact the unit group of this ring is infinite.

In fact, Dirichlet's unit theorem describes the structure of U(R) precisely: it is isomorphic to a group of the form

where is the (finite, cyclic) group of roots of unity in R and n, the rank of the unit group is

where are the numbers of real embeddings and the number of pairs of complex embeddings of F, respectively.

This recovers the above example: the unit group of (the ring of integers of) a real quadratic field is infinite of rank 1, since .

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.

Polynomials and power series[edit]

For a commutative ring R, the units of the polynomial ring R[x] are precisely those polynomials

such that is a unit in R, and the remaining coefficients are nilpotent elements, i.e., satisfy for some N.[4] In particular, if R is a domain (has no zero divisors), then the units of R[x] agree with the ones of R. The units of the power series ring are precisely those power series

such that is a unit in R.[5]

Matrix rings[edit]

The unit group of the ring Mn(F) of n × n matrices over a field F is the group GLn(F) of invertible matrices.

Group of units[edit]

The units of a ring R form a group U(R) under multiplication, the group of units of R. Other common notations for U(R) are R, R×, and E(R) (from the German term Einheit).

This defins is a functor U from the category of rings to the category of groups: every ring homomorphism f : RS 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.

Associatedness[edit]

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

rs

means that there is a unit u with r = us.

In an integral domain the cardinality of an equivalence class of associates is the same as that of U(R).

See also[edit]

References[edit]

  1. ^ Dummit, David S.; Foote, Richard M. (2004). Abstract Algebra (3rd ed.). John Wiley & Sons. ISBN 0-471-43334-9. 
  2. ^ Lang, Serge (2002). Algebra. Graduate Texts in Mathematics. Springer. ISBN 0-387-95385-X. 
  3. ^ In a ring, the additive inverse of a non-zero element can equal to the element itself.
  4. ^ Watkins (2007, Theorem 11.1)
  5. ^ Watkins (2007, Theorem 12.1)