Коммутативті сақиналар және коммутативті алгебра

Algebraic structure
In mathematics, 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 ring properties that are not specific to commutative rings. This distinction results from the high number of fundamental properties of commutative rings that do not extend to noncommutative rings.

A ring is a set 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. and 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., The identity elements for addition and multiplication are denoted and , respectively. If the multiplication is commutative, i. e.

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

An important example, and in some sense crucial, is the ring of integers 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 as an abbreviation of the German word Zahlen (numbers). A field is a commutative ring where and every non zero element is invertible; i. e., has a multiplicative inverse such that Therefore, by definition, any field is a commutative ring. The rational, real and complex numbers form fields. If is a given commutative ring, then the set of all polynomials in the variable whose coefficients are in forms the polynomial ring, denoted The same holds true for several variables. If is some topological space, for example a subset of some , real or complex valued continuous functions on form a commutative ring. The same is true for differentiable or holomorphic functions, when the two concepts are defined, such as for a complex manifold.

In contrast to fields, where every nonzero element is multiplicatively invertible, the concept of divisibility for rings is richer. An element of ring is called a unit if it possesses a multiplicative inverse. Another particular type of element is the zero divisors, i. e. an element such that there exists a non zero element of the ring such that If possesses no non zero zero divisors, it is called an integral domain (or domain). An element satisfying for some positive integer is called nilpotent.

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.

For a ring , an module is like what a vector space is to a field. That is, elements in a module can be added; they can be multiplied by elements of subject to the same axioms as for a vector space. The study of modules is significantly more involved than the one of vector spaces, since there are modules that do not have any basis, that is, do not contain a spanning set whose elements are linearly independents. A module that has a basis is called a free module, and a submodule of a free module needs not to be free. A module of finite type is a module that has a finite spanning set. Modules of finite type play a fundamental role in the theory of commutative rings, similar to the role of the finite dimensional vector spaces in linear algebra. In particular, Noetherian rings (see also , below) can be defined as the rings such that every submodule of a module of finite type is also of finite type.

Ideals of a ring are the submodules of , i. e., the modules contained in In more detail, an ideal is a non empty subset of such that for all in , and in , both and are in For various applications, understanding the ideals of a ring is of particular importance, but often one proceeds by studying modules in general. Any ring has two ideals, namely the zero ideal and , the whole ring. These two ideals are the only ones precisely if is a field. Given any subset of (where is some index set), the ideal generated by is the smallest ideal that contains Equivalently, it is given by finite linear combinations

If consists of a single element , the ideal generated by consists of the multiples of , i. e., the elements of the form for arbitrary elements Such an ideal is called a principal ideal. If every ideal is a principal ideal, is called a principal ideal ring; two important cases are and , the polynomial ring over a field These two are in addition domains, so they are called principal ideal domains. Unlike for general rings, for a principal ideal domain, the properties of individual elements are strongly tied to the properties of the ring as a whole. For example, any principal ideal domain is a unique factorization domain (UFD) which means that any element is a product of irreducible elements, in a (up to reordering of factors) unique way. Here, an element in a domain is called irreducible if the only way of expressing it as a product

is by either or being a unit. An example, important in field theory, are irreducible polynomials, i. e., irreducible elements in , for a field The fact that is a UFD can be stated more elementarily by saying that any natural number can be uniquely decomposed as product of powers of prime numbers. It is also known as the fundamental theorem of arithmetic. An element is a prime element if whenever divides a product , divides or In a domain, being prime implies being irreducible. The converse is true in a unique factorization domain, but false in general.

The spectrum of a ring , denoted by , is the set of all prime ideals of It is equipped with a topology, the Zariski topology, which reflects the algebraic properties of : a basis of open subsets is given by
where is any ring element. Interpreting as a function that takes the value f mod p (i. e., the image of f in the residue field R/p), this subset is the locus where f is non zero. The spectrum also makes precise the intuition that localisation and factor rings are complementary: the natural maps R → Rf 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. Even for basic rings, such as illustrated for 1=R = Z at the right, the Zariski topology is quite different from the one on the set of real numbers. 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
1= {x =(x1, , xn) ∊ kn | f1(x) = = fm(x) = 0.} 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 notion of a spectrum is the common basis of commutative algebra and algebraic geometry. Algebraic geometry proceeds by endowing Spec R with a sheaf (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 Moreover, this 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
1= 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 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.

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
1= R[T1, , Tn] / I. 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.

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.

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. lists some open questions in this area of active research.

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 1=rn = r. If, 1=r2 = r for every r, the ring is called Boolean ring. More general conditions which guarantee commutativity of a ring are also known.

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.