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. Complementarily, noncommutative algebra is the study of noncommutative rings where multiplication is not required to be commutative.

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.

The spectrum contains the set of maximal ideals, which is occasionally denoted mSpec (R). For an algebraically closed field k, mSpec (k[T1, ..., Tn] / (f1, ..., fm)) is in bijection with the set

Thus, maximal ideals reflect the geometric properties of solution sets of polynomials, which is an initial motivation for the study of commutative rings. However, the consideration of non-maximal ideals as part of the geometric properties of a ring is useful for several reasons. For example, the minimal prime ideals (i.e., the ones not strictly containing smaller ones) correspond to the irreducible components of Spec R. For a Noetherian ring R, Spec R has only finitely many irreducible components. This is a geometric restatement of primary decomposition, according to which any ideal can be decomposed as a product of finitely many primary ideals. This fact is the ultimate generalization of the decomposition into prime ideals in Dedekind rings.

The resulting equivalence of the two said categories aptly reflects algebraic properties of rings in a geometrical manner.

Similar to the fact that manifolds are locally given by open subsets of Rn, affine schemes are local models for schemes, which are the object of study in algebraic geometry. Therefore, several notions concerning commutative rings stem from geometric intuition.

The Krull dimension (or dimension) dim R of a ring R measures the "size" of a ring by, roughly speaking, counting independent elements in R. The dimension of algebras over a field k can be axiomatized by four properties:

The dimension is defined, for any ring R, as the supremum of lengths n of chains of prime ideals

For example, a field is zero-dimensional, since the only prime ideal is the zero ideal. The integers are one-dimensional, since chains are of the form (0) ⊊ (p), where p is a prime number. For non-Noetherian rings, and also non-local rings, the dimension may be infinite, but Noetherian local rings have finite dimension. Among the four axioms above, the first two are elementary consequences of the definition, whereas the remaining two hinge on important facts in commutative algebra, the going-up theorem and Krull's principal ideal theorem.

A ring homomorphism or, more colloquially, simply a map, is a map f : RS such that

These conditions ensure f(0) = 0. Similarly as for other algebraic structures, a ring homomorphism is thus a map that is compatible with the structure of the algebraic objects in question. 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

A ring homomorphism is called an isomorphism if it is bijective. An example of a ring isomorphism, known as the Chinese remainder theorem, is

Commutative rings, together with ring homomorphisms, form a category. The ring Z is the initial object in this category, which means that for any commutative ring R, there is a unique ring homomorphism ZR. By means of this map, an integer n can be regarded as an element of R. For example, the binomial formula

is again a commutative R-algebra. In some cases, the tensor product can serve to find a T-algebra which relates to Z as S relates to R. For example,

An R-algebra S is called finitely generated (as an algebra) if there are finitely many elements s1, ..., sn such that any element of s is expressible as a polynomial in the si. Equivalently, S is isomorphic to

A much stronger condition is that S is finitely generated as an R-module, which means that any s can be expressed as a R-linear combination of some finite set s1, ..., sn.

A ring is called local if it has only a single maximal ideal, denoted by m. For any (not necessarily local) ring R, the localization

at a prime ideal p is local. This localization reflects the geometric properties of Spec R "around p". Several notions and problems in commutative algebra can be reduced to the case when R is local, making local rings a particularly deeply studied class of rings. The residue field of R is defined as

Any R-module M yields a k-vector space given by M / mM. Nakayama's lemma shows this passage is preserving important information: a finitely generated module M is zero if and only if M / mM is zero.

The k-vector space m/m2 is an algebraic incarnation of the cotangent space. Informally, the elements of m can be thought of as functions which vanish at the point p, whereas m2 contains the ones which vanish with order at least 2. For any Noetherian local ring R, the inequality

holds true, reflecting the idea that the cotangent (or equivalently the tangent) space has at least the dimension of the space Spec R. If equality holds true in this estimate, R is called a regular local ring. A Noetherian local ring is regular if and only if the ring (which is the ring of functions on the tangent cone)

Discrete valuation rings are equipped with a function which assign an integer to any element r. This number, called the valuation of r can be informally thought of as a zero or pole order of r. Discrete valuation rings are precisely the one-dimensional regular local rings. For example, the ring of germs of holomorphic functions on a Riemann surface is a discrete valuation ring.

The twisted cubic (green) is a set-theoretic complete intersection, but not a complete intersection.

By Krull's principal ideal theorem, a foundational result in the dimension theory of rings, the dimension of

is at least rn. A ring R is called a complete intersection ring if it can be presented in a way that attains this minimal bound. This notion is also mostly studied for local rings. Any regular local ring is a complete intersection ring, but not conversely.

A ring R is a set-theoretic complete intersection if the reduced ring associated to R, i.e., the one obtained by dividing out all nilpotent elements, is a complete intersection. As of 2017, it is in general unknown, whether curves in three-dimensional space are set-theoretic complete intersections.[3]

The depth of a local ring R is the number of elements in some (or, as can be shown, any) maximal regular sequence, i.e., a sequence a1, ..., anm such that all ai are non-zero divisors in

holds. A local ring in which equality takes place is called a Cohen–Macaulay ring. Local complete intersection rings, and a fortiori, regular local rings are Cohen–Macaulay, but not conversely. Cohen–Macaulay combine desirable properties of regular rings (such as the property of being universally catenary rings, which means that the (co)dimension of primes is well-behaved), but are also more robust under taking quotients than regular local rings.[4]

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[clarification needed] a ring normal is known as normalization.

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. This ring serves as an algebraic analogue of the disk. Analogously, the ring of p-adic integers is the completion of Z with respect to the principal ideal (p). Any ring that is isomorphic to its own completion, is called complete.

Complete local rings satisfy Hensel's lemma, which roughly speaking allows extending solutions (of various problems) over the residue field k to R.

Several deeper aspects of commutative rings have been studied using methods from homological algebra. Hochster (2007) lists some open questions in this area of active research.

Projective modules can be defined to be the direct summands of free modules. If R is local, any finitely generated projective module is actually free, which gives content to an analogy between projective modules and vector bundles.[5] The Quillen–Suslin theorem asserts that any finitely generated projective module over k[T1, ..., Tn] (k a field) is free, but in general these two concepts differ. A local Noetherian ring is regular if and only if its global dimension is finite, say n, which means that any finitely generated R-module has a resolution by projective modules of length at most n.

The proof of this and other related statements relies on the usage of homological methods, such as the Ext functor. This functor is the derived functor of the functor

The latter functor is exact if M is projective, but not otherwise: for a surjective map EF of R-modules, a map MF need not extend to a map ME. The higher Ext functors measure the non-exactness of the Hom-functor. The importance of this standard construction in homological algebra stems can be seen from the fact that a local Noetherian ring R with residue field k is regular if and only if

vanishes for all large enough n. Moreover, the dimensions of these Ext-groups, known as Betti numbers, grow polynomially in n if and only if R is a local complete intersection ring.[6] A key argument in such considerations is the Koszul complex, which provides an explicit free resolution of the residue field k of a local ring R in terms of a regular sequence.

The tensor product is another non-exact functor relevant in the context of commutative rings: for a general R-module M, the functor

is only right exact. If it is exact, M is called flat. If R is local, any finitely presented flat module is free of finite rank, thus projective. Despite being defined in terms of homological algebra, flatness has profound geometric implications. For example, if an R-algebra S is flat, the dimensions of the fibers

(for prime ideals p in R) have the "expected" dimension, namely dim S − dim R + dim (R / p).

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.[7] If, r2 = r for every r, the ring is called Boolean ring. More general conditions which guarantee commutativity of a ring are also known.[8]

If the Ri are connected by differentials ∂ such that an abstract form of the product rule holds, i.e.,

R is called a commutative differential graded algebra (cdga). An example is the complex of differential forms on a manifold, with the multiplication given by the exterior product, is a cdga. The cohomology of a cdga is a graded-commutative ring, sometimes referred to as the cohomology ring. A broad range examples of graded rings arises in this way. For example, the Lazard ring is the ring of cobordism classes of complex manifolds.

A graded-commutative ring with respect to a grading by Z/2 (as opposed to Z) is called a superalgebra.

A related notion is an almost commutative ring, which means that R is filtered in such a way that the associated graded ring

is commutative. An example is the Weyl algebra and more general rings of differential operators.

A simplicial commutative ring is a simplicial object in the category of commutative rings. They are building blocks for (connective) derived algebraic geometry. A closely related but more general notion is that of E-ring.