Commutative ring

In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra.

Some specific kinds of commutative rings are given with the following chain of class inclusions:

commutative rings ⊃ integral domains ⊃ integrally closed domains ⊃ GCD domains ⊃ unique factorization domains ⊃ principal ideal domains ⊃ Euclidean domains ⊃ fields ⊃ finite fields

Definition and first examples

Definition

A ring is a set R equipped with two binary operations, i.e. operations combining any two elements of the ring to a third. They are called addition and multiplication and commonly denoted by "+" and "⋅"; e.g. a + b and a ⋅ b. To form a ring these two operations have to satisfy a number of properties: the ring has to be an abelian group under addition as well as a monoid under multiplication, where multiplication distributes over addition; i.e., a ⋅ (b + c) = (a ⋅ b) + (a ⋅ c). The identity elements for addition and multiplication are denoted 0 and 1, respectively.

If the multiplication is commutative, i.e.

a ⋅ b = b ⋅ a,

then the ring R is called commutative. In the remainder of this article, all rings will be commutative, unless explicitly stated otherwise.

First examples

An important example, and in some sense crucial, is the ring of integers Z with the two operations of addition and multiplication. As the multiplication of integers is a commutative operation, this is a commutative ring. It is usually denoted Z as an abbreviation of the German word Zahlen (numbers).

A field is a commutative ring where every non-zero element a is invertible; i.e., has a multiplicative inverse b such that a ⋅ b = 1. Therefore, by definition, any field is a commutative ring. The rational, real and complex numbers form fields.

The ring of 2×2 matrices is not commutative, since matrix multiplication fails to be commutative, as the following example shows:

$\begin{align} \begin{bmatrix} 1 & 1\\ 0 & 1\\ \end{bmatrix}\cdot \begin{bmatrix} 1 & 1\\ 1 & 0\\ \end{bmatrix} &= \begin{bmatrix} 2 & 1\\ 1 & 0\\ \end{bmatrix}\\ \begin{bmatrix} 1 & 1\\ 1 & 0\\ \end{bmatrix}\cdot \begin{bmatrix} 1 & 1\\ 0 & 1\\ \end{bmatrix} &= \begin{bmatrix} 1 & 2\\ 1 & 1\\ \end{bmatrix} \end{align}$

However, matrices that can be diagonalized with the same similarity transformation do form a commutative ring. An example is the set of matrices of divided differences with respect to a fixed set of nodes.

If R is a given commutative ring, then the set of all polynomials in the variable X whose coefficients are in R forms the polynomial ring, denoted R[X]. The same holds true for several variables.

If V is some topological space, for example a subset of some Rⁿ, real- or complex-valued continuous functions on V form a commutative ring. The same is true for differentiable or holomorphic functions, when the two concepts are defined, such as for V a complex manifold.

Ideals and the spectrum

In contrast to fields, where every nonzero element is multiplicatively invertible, the theory of rings is more complicated. There are several notions to cope with that situation. First, an element a of ring R is called a unit if it possesses a multiplicative inverse. Another particular type of element is the zero divisors, i.e. a non-zero element a such that there exists a non-zero element b of the ring such that ab = 0. If R possesses no zero divisors, it is called an integral domain since it closely resembles the integers in some ways.

Many of the following notions also exist for not necessarily commutative rings, but the definitions and properties are usually more complicated. For example, all ideals in a commutative ring are automatically two-sided, which simplifies the situation considerably.

Ideals and factor rings

The inner structure of a commutative ring is determined by considering its ideals, i.e. nonempty subsets that are closed under multiplication with arbitrary ring elements and addition: for all r in R, i and j in I, both ri and i + j are required to be in I. Given any subset F = {f_j}_{j ∈ J} of R (where J is some index set), the ideal generated by F is the smallest ideal that contains F. Equivalently, it is given by finite linear combinations

r₁f₁ + r₂f₂ + ... + r_nf_n.

An ideal generated by one element is called a principal ideal. A ring all of whose ideals are principal is called a principal ideal ring; two important cases are Z and k[X], the polynomial ring over a field k. Any ring has two ideals, namely the zero ideal {0} and R, the whole ring. An ideal is proper if it is strictly smaller than the whole ring. An ideal that is not strictly contained in any proper ideal is called maximal. An ideal m is maximal if and only if R / m is a field. Except for the zero ring, any ring (with identity) possesses at least one maximal ideal; this follows from Zorn's lemma.

The definition of ideals is such that "dividing" I "out" gives another ring, the factor ring R / I: it is the set of cosets of I together with the operations

(a + I) + (b + I) = (a + b) + I and (a + I)(b + I) = ab + I.

For example, the ring Z/nZ (also denoted Z_n), where n is an integer, is the ring of integers modulo n. It is the basis of modular arithmetic.

Localizations

The localization of a ring is the counterpart to factor rings insofar as in a factor ring R / I certain elements (namely the elements of I) become zero, whereas in the localization certain elements are rendered invertible, i.e. multiplicative inverses are added to the ring. Concretely, if S is a multiplicatively closed subset of R (i.e. whenever s, t ∈ S then so is st) then the localization of R at S, or ring of fractions with denominators in S, usually denoted S⁻¹R consists of symbols

$\frac{r}{s}$ with r ∈ R, s ∈ S

subject to certain rules that mimick the cancellation familiar from rational numbers. Indeed, in this language Q is the localization of Z at all nonzero integers. This construction works for any integral domain R instead of Z. The localization (R \ {0})⁻¹R is called the quotient field of R. If S consists of the powers of one fixed element f, the localisation is written R_f.

Prime ideals and the spectrum

A particularly important type of ideals is prime ideals, often denoted p. This notion arose when algebraists (in the 19th century) realized that, unlike in Z, in many rings there is no unique factorization into prime numbers. (Rings where it does hold are called unique factorization domains.) By definition, a prime ideal is a proper ideal such that, whenever the product ab of any two ring elements a and b is in p, at least one of the two elements is already in p. (The opposite conclusion holds for any ideal, by definition). Equivalently, the factor ring R / p is an integral domain. Yet another way of expressing the same is to say that the complement R \ p is multiplicatively closed. The localisation (R \ p)⁻¹R is important enough to have its own notation: R_p. This ring has only one maximal ideal, namely pR_p. Such rings are called local.

By the above, any maximal ideal is prime. Proving that an ideal is prime, or equivalently that a ring has no zero-divisors can be very difficult.

The spectrum of Z.

Prime ideals are the key step in interpreting a ring geometrically, via the spectrum of a ring Spec R: it is the set of all prime ideals of R.^{[nb 1]} As noted above, there is at least one prime ideal, therefore the spectrum is nonempty. If R is a field, the only prime ideal is the zero ideal, therefore the spectrum is just one point. The spectrum of Z, however, contains one point for the zero ideal, and a point for any prime number p (which generates the prime ideal pZ). The spectrum is endowed with a topology called the Zariski topology, which is determined by specifying that subsets D(f) = {p ∈ Spec R, f ∉ p}, where f is any ring element, be open. This topology tends to be different from those encountered in analysis or differential geometry; for example, there will generally be points which are not closed. The closure of the point corresponding to the zero ideal 0 ⊂ Z, for example, is the whole spectrum of Z.

The notion of a spectrum is the common basis of commutative algebra and algebraic geometry. Algebraic geometry proceeds by endowing Spec R with a sheaf $\mathcal O$ (an entity that collects functions defined locally, i.e. on varying open subsets). The datum of the space and the sheaf is called an affine scheme. Given an affine scheme, the underlying ring R can be recovered as the global sections of $\mathcal O$. Moreover, the established one-to-one correspondence between rings and affine schemes is also compatible with ring homomorphisms: any f : R → S gives rise to a continuous map in the opposite direction

Spec S → Spec R, q ↦ f⁻¹(q), i.e. any prime ideal of S is mapped to its preimage under f, which is a prime ideal of R.

The spectrum also makes precise the intuition that localisation and factor rings are complementary: the natural maps R → R_f and R → R / fR correspond, after endowing the spectra of the rings in question with their Zariski topology, to complementary open and closed immersions respectively.

Altogether the equivalence of the two said categories is very apt to reflect algebraic properties of rings in a geometrical manner. Affine schemes are–much the same way as manifolds are locally given by open subsets of Rⁿ–local models for schemes, which are the object of study in algebraic geometry. Therefore, many notions that apply to rings and homomorphisms stem from geometric intuition.

Ring homomorphisms

As usual in algebra, a function f between two objects that respects the structures of the objects in question is called homomorphism. In the case of rings, a ring homomorphism is a map f : R → S such that

f(a + b) = f(a) + f(b), f(ab) = f(a)f(b) and f(1) = 1.

These conditions ensure f(0) = 0, but the requirement that the multiplicative identity element 1 is preserved under f would not follow from the two remaining properties. In such a situation S is also called an R-algebra, by understanding that s in S may be multiplied by some r of R, by setting

r · s := f(r) · s.

The kernel and image of f are defined by ker (f) = {r ∈ R, f(r) = 0} and im (f) = f(R) = {f(r), r ∈ R}. The kernel is an ideal of R, and the image is a subring of S.

Modules

The outer structure of a commutative ring is determined by considering linear algebra over that ring, i.e., by investigating the theory of its modules, which are similar to vector spaces, except that the base is not necessarily a field, but can be any ring R. The theory of R-modules is significantly more difficult than linear algebra of vector spaces. Module theory has to grapple with difficulties such as modules not having bases, that the rank of a free module (i.e. the analog of the dimension of vector spaces) may not be well-defined and that submodules of finitely generated modules need not be finitely generated (unless R is Noetherian, see below).

Ideals within a ring R can be characterized as R-modules which are submodules of R. On the one hand, a good understanding of R-modules necessitates enough information about R. Vice versa, however, many techniques in commutative algebra that study the structure of R, by examining its ideals, proceed by studying modules in general.

Noetherian rings

A ring is called Noetherian (in honor of Emmy Noether, who developed this concept) if every ascending chain of ideals

0 ⊆ I₀ ⊆ I₁ ... ⊆ I_n ⊆ I_{n + 1} ⊆ ...

becomes stationary, i.e. becomes constant beyond some index n. Equivalently, any ideal is generated by finitely many elements, or, yet equivalent, submodules of finitely generated modules are finitely generated. A ring is called Artinian (after Emil Artin), if every descending chain of ideals

R ⊇ I₀ ⊇ I₁ ... ⊇ I_n ⊇ I_{n + 1} ⊇ ...

becomes stationary eventually. Despite the two conditions appearing symmetric, Noetherian rings are much more general than Artinian rings. For example, Z is Noetherian, since every ideal can be generated by one element, but is not Artinian, as the chain

Z ⊋ 2Z ⊋ 4Z ⊋ 8Z ⊋ ...

shows. In fact, by the Hopkins–Levitzki theorem, every Artinian ring is Noetherian.

Being Noetherian is an extremely important finiteness condition, and the condition is preserved under many operations that occur frequently in geometry. For example, if R is Noetherian, then so is:

• the polynomial ring R[X₁, X₂, ..., X_n] (by Hilbert's basis theorem);
• any localization S⁻¹R;
• any factor ring R / I.

Dimension

The Krull dimension (or simply dimension) dim R of a ring R is a notion to measure the "size" of a ring, very roughly by the counting independent elements in R. Precisely, it is defined as the supremum of lengths n of chains of prime ideals

$\mathfrak{p}_0\subsetneq \mathfrak{p}_1\subsetneq \ldots \subsetneq \mathfrak{p}_n$.

For example, a field is zero-dimensional, since the only prime ideal is the zero ideal. It is also known that a commutative ring is Artinian if and only if it is Noetherian and zero-dimensional (i.e., all its prime ideals are maximal). The integers are one-dimensional: any chain of prime ideals is of the form

$0 = \mathfrak p_0 \subsetneq p\mathbb Z = \mathfrak p_1$, where p is a prime number

since any ideal in Z is principal.

The dimension behaves well if the rings in question are Noetherian: the expected equality

dim R[X] = dim R + 1

holds in this case (in general, one has only dim R + 1 ≤ dim R[X] ≤ 2 · dim R + 1). Furthermore, since the dimension depends only on one maximal chain, the dimension of R is the supremum of all dimensions of its localisations R_p, where p is an arbitrary prime ideal. Intuitively, the dimension of R is a local property of the spectrum of R. Therefore, the dimension is often considered for local rings only, also since general Noetherian rings may still be infinite, despite all their localisations being finite-dimensional.

Determining the dimension of, say,

k[X₁, X₂, ..., X_n] / (f₁, f₂, ..., f_m), where k is a field and the f_i are some polynomials in n variables,

is generally not easy. For R Noetherian, the dimension of R / I is, by Krull's principal ideal theorem, at least dim R − n, if I is generated by n elements. If the dimension does drops as much as possible, i.e. dim R / I = dim R − n, the R / I is called a complete intersection.

A local ring R, i.e. one with only one maximal ideal m, is called regular, if the (Krull) dimension of R equals the dimension (as a vector space over the field R / m) of the cotangent space m / m².

Constructing commutative rings

There are several ways to construct new rings out of given ones. The aim of such constructions is often to improve certain properties of the ring so as to make it more readily understandable. For example, an integral domain that is integrally closed in its field of fractions is called normal. This is a desirable property, for example any normal one-dimensional ring is necessarily regular. Rendering a ring normal is known as normalization.

Completions

If I is an ideal in a commutative ring R, the powers of I form topological neighborhoods of 0 which allow R to be viewed as a topological ring. This topology is called the I-adic topology. R can then be completed with respect to this topology. Formally, the I-adic completion is the inverse limit of the rings R/Iⁿ. For example, if k is a field, k[[X]], the formal power series ring in one variable over k, is the I-adic completion of k[X] where I is the principal ideal generated by X. Analogously, the ring of p-adic integers is the I-adic completion of Z where I is the principal ideal generated by p. Any ring that is isomorphic to its own completion, is called complete.

Properties

By Wedderburn's theorem, every finite division ring is commutative, and therefore a finite field. Another condition ensuring commutativity of a ring, due to Jacobson, is the following: for every element r of R there exists an integer n > 1 such that rⁿ = r.^[1] If, r² = r for every r, the ring is called Boolean ring. More general conditions which guarantee commutativity of a ring are also known.^[2]

See also

• Graded ring
• Almost commutative ring
• Almost ring, a certain generalization of a commutative ring.
• Simplicial commutative ring, a simplicial object in the category of commutative rings.

Notes

1. This notion can be related to the spectrum of a linear operator, see Spectrum of a C*-algebra and Gelfand representation.

Citations

1. Jacobson 1945
2. Pinter-Lucke 2007

References

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