Unit (ring theory)

Not to be confused with Unit ring.

In algebra, a unit or invertible element ^{[lower-alpha 1]} of a ring is an invertible element for the multiplication of the ring. That is, an element u of a ring R is a unit if there exists v in R such that $${\displaystyle vu=uv=1,}$$ where 1 is the multiplicative identity; the element v is unique for this property and is called the multiplicative inverse of u. ^{[1] [2]} The set of units of R forms a group R^× under multiplication, called the group of units or unit group of R. ^{[lower-alpha 2]} Other notations for the unit group are R^∗, U(R), and E(R) (from the German term Einheit ).

Less commonly, the term unit is sometimes used to refer to the element 1 of the ring, in expressions like ring with a unit or unit ring, and also unit matrix. Because of this ambiguity, 1 is more commonly called the "unity" or the "identity" of the ring, and the phrases "ring with unity" or a "ring with identity" may be used to emphasize that one is considering a ring instead of a rng.

Examples

The multiplicative identity 1 and its additive inverse −1 are always units. 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. In a nonzero ring, the element 0 is not a unit, so R^× is not closed under addition. A nonzero ring R in which every nonzero element is a unit (that is, R^× = R∖ {0}) is called a division ring (or a skew-field). A commutative division ring is called a field. For example, the unit group of the field of real numbers R is R∖ {0}.

Integer ring

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.

Ring of integers of a number field

In the ring Z[√3] obtained by adjoining the quadratic integer √3 to Z, one has (2 + √3)(2 − √3) = 1, so 2 + √3 is a unit, and so are its powers, so Z[√3] has infinitely many units.

More generally, for the ring of integers R in a number field F, Dirichlet's unit theorem states that R^× is isomorphic to the group $${\displaystyle \mathbf {Z} ^{n}\times \mu _{R}}$$ where ${\displaystyle \mu _{R}}$ is the (finite, cyclic) group of roots of unity in R and n, the rank of the unit group, is $${\displaystyle n=r_{1}+r_{2}-1,}$$ where ${\displaystyle r_{1},r_{2}}$ are the number of real embeddings and the number of pairs of complex embeddings of F, respectively.

This recovers the Z[√3] example: The unit group of (the ring of integers of) a real quadratic field is infinite of rank 1, since ${\displaystyle r_{1}=2,r_{2}=0}$.

Polynomials and power series

For a commutative ring R, the units of the polynomial ring R[x] are the polynomials $${\displaystyle p(x)=a_{0}+a_{1}x+\dots +a_{n}x^{n}}$$ such that a₀ is a unit in R and the remaining coefficients ${\displaystyle a_{1},\dots ,a_{n}}$ are nilpotent, i.e., satisfy ${\displaystyle a_{i}^{N}=0}$ for some N. ^[4] In particular, if R is a domain (or more generally reduced), then the units of R[x] are the units of R. The units of the power series ring ${\displaystyle R[[x]]}$ are the power series $${\displaystyle p(x)=\sum _{i=0}^{\infty }a_{i}x^{i}}$$ such that a₀ is a unit in R. ^[5]

Matrix rings

The unit group of the ring M_n(R) of n × n matrices over a ring R is the group GL_n(R) of invertible matrices. For a commutative ring R, an element A of M_n(R) is invertible if and only if the determinant of A is invertible in R. In that case, A⁻¹ can be given explicitly in terms of the adjugate matrix.

In general

For elements x and y in a ring R, if ${\displaystyle 1-xy}$ is invertible, then ${\displaystyle 1-yx}$ is invertible with inverse ${\displaystyle 1+y(1-xy)^{-1}x}$; ^[6] this formula can be guessed, but not proved, by the following calculation in a ring of noncommutative power series: $${\displaystyle (1-yx)^{-1}=\sum _{n\geq 0}(yx)^{n}=1+y\left(\sum _{n\geq 0}(xy)^{n}\right)x=1+y(1-xy)^{-1}x.}$$ See Hua's identity for similar results.

Group of units

A commutative ring is a local ring if R∖R^× is a maximal ideal.

As it turns out, if R∖R^× is an ideal, then it is necessarily a maximal ideal and R is local since a maximal ideal is disjoint from R^×.

If R is a finite field, then R^× is a cyclic group of order |R| − 1.

Every ring homomorphism f : R → S induces a group homomorphism R^× → S^×, since f maps units to units. In fact, the formation of the unit group defines a functor from the category of rings to the category of groups. This functor has a left adjoint which is the integral group ring construction. ^[7]

The group scheme ${\displaystyle \operatorname {GL} _{1}}$ is isomorphic to the multiplicative group scheme ${\displaystyle \mathbb {G} _{m}}$ over any base, so for any commutative ring R, the groups ${\displaystyle \operatorname {GL} _{1}(R)}$ and ${\displaystyle \mathbb {G} _{m}(R)}$ are canonically isomorphic to U(R). Note that the functor ${\displaystyle \mathbb {G} _{m}}$ (that is, R ↦ U(R)) is representable in the sense: ${\displaystyle \mathbb {G} _{m}(R)\simeq \operatorname {Hom} (\mathbb {Z} [t,t^{-1}],R)}$ for commutative rings R (this for instance follows from the aforementioned adjoint relation with the group ring construction). Explicitly this means that there is a natural bijection between the set of the ring homomorphisms ${\displaystyle \mathbb {Z} [t,t^{-1}]\to R}$ and the set of unit elements of R (in contrast, ${\displaystyle \mathbb {Z} [t]}$ represents the additive group ${\displaystyle \mathbb {G} _{a}}$, the forgetful functor from the category of commutative rings to the category of abelian groups).

Associatedness

Suppose that R is commutative. Elements r and s of R are called associate if there exists a unit u in R such that r = us; then write r ~ s. In any ring, pairs of additive inverse elements ^{[lower-alpha 3]} x and −x are associate, since any ring includes the unit −1. For example, 6 and −6 are associate in Z. In general, ~ is an equivalence relation on R.

Associatedness can also be described in terms of the action of R^× on R via multiplication: Two elements of R are associate if they are in the same R^×-orbit.

In an integral domain, the set of associates of a given nonzero element has the same cardinality as R^×.

The equivalence relation ~ can be viewed as any one of Green's semigroup relations specialized to the multiplicative semigroup of a commutative ring R.

See also

• S-units
• Localization of a ring and a module

Notes

1. In the case of rings, the use of "invertible element" is taken as self-evidently referring to multiplication, since all elements of a ring are invertible for addition.
2. The notation R^×, introduced by André Weil, is commonly used in number theory, where unit groups arise frequently. ^[3] The symbol × is a reminder that the group operation is multiplication. Also, a superscript × is not frequently used in other contexts, whereas a superscript * often denotes dual.
3. x and −x are not necessarily distinct. For example, in the ring of integers modulo 6, one has 3 = −3 even though 1 ≠ −1.

Citations

1. Dummit & Foote 2004
2. Lang 2002
3. Weil 1974
4. Watkins 2007, Theorem 11.1
5. Watkins 2007, Theorem 12.1
6. Jacobson 2009, §2.2 Exercise 4
7. Cohn 2003, §2.2 Exercise 10

Sources

• Cohn, Paul M. (2003). Further algebra and applications (Revised ed. of Algebra, 2nd ed.). London: Springer-Verlag. ISBN   1-85233-667-6. Zbl   1006.00001.
• Dummit, David S.; Foote, Richard M. (2004). Abstract Algebra (3rd ed.). John Wiley & Sons. ISBN   0-471-43334-9.
• Jacobson, Nathan (2009). Basic Algebra 1 (2nd ed.). Dover. ISBN   978-0-486-47189-1.
• Lang, Serge (2002). Algebra. Graduate Texts in Mathematics. Springer. ISBN   0-387-95385-X.
• Watkins, John J. (2007), Topics in commutative ring theory, Princeton University Press, ISBN   978-0-691-12748-4, MR   2330411
• Weil, André (1974). Basic number theory. Grundlehren der mathematischen Wissenschaften. Vol. 144 (3rd ed.). Springer-Verlag. ISBN   978-3-540-58655-5.