Manual Ring Theory: 001

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Some authors define a ring without the requirement of associativity for multiplication. Then Z 4 is a ring: each axiom follows from the corresponding axiom for Z. The set of 2-by-2 matrices with real number entries is written. With the operations of matrix addition and matrix multiplication , this set satisfies the above ring axioms. More generally, for any ring R , commutative or not, and any nonnegative integer n , one may form the ring of n -by- n matrices with entries in R : see Matrix ring.

The study of rings originated from the theory of polynomial rings and the theory of algebraic integers. But Dedekind did not use the term "ring" and did not define the concept of a ring in a general setting. The term "Zahlring" number ring was coined by David Hilbert in and published in According to Harvey Cohn, Hilbert used the term for a ring that had the property of "circling directly back" to an element of itself. The first axiomatic definition of a ring was given by Adolf Fraenkel in , [13] [14] but his axioms were stricter than those in the modern definition.

For instance, he required every non-zero-divisor to have a multiplicative inverse. Fraenkel required a ring to have a multiplicative identity 1, [17] whereas Noether did not. Most or all books on algebra [18] [19] up to around followed Noether's convention of not requiring a 1. Starting in the s, it became increasingly common to see books including the existence of 1 in the definition of ring, especially in advanced books by notable authors such as Artin, [20] Atiyah and MacDonald, [21] Bourbaki, [22] Eisenbud, [23] and Lang.

Faced with this terminological ambiguity, some authors have tried to impose their views, while others have tried to adopt more precise terms. In the first category, we find for instance Gardner and Wiegandt, who argue that if one requires all rings to have a 1, then some consequences include the lack of existence of infinite direct sums of rings, and the fact that proper direct summands of rings are not subrings. They conclude that "in many, maybe most, branches of ring theory the requirement of the existence of a unity element is not sensible, and therefore unacceptable.

In the second category, we find authors who use the following terms: [30] [31]. One example of a nilpotent element is a nilpotent matrix. A nilpotent element in a nonzero ring is necessarily a zero divisor. One example of an idempotent element is a projection in linear algebra. A subset S of R is said to be a subring if it can be regarded as a ring with the addition and the multiplication restricted from R to S.

If all rings have been assumed, by convention, to have a multiplicative identity, then to be a subring one would also require S to share the same identity element as R.


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For example, the ring Z of integers is a subring of the field of real numbers and also a subring of the ring of polynomials Z [ X ] in both cases, Z contains 1, which is the multiplicative identity of the larger rings. On the other hand, the subset of even integers 2 Z does not contain the identity element 1 and thus does not qualify as a subring of Z. An intersection of subrings is a subring. The smallest subring containing a given subset E of R is called a subring generated by E.

Such a subring exists since it is the intersection of all subrings containing E. For a ring R , the smallest subring containing 1 is called the characteristic subring of R. If n is the smallest positive integer such that this occurs, then n is called the characteristic of R. More generally, given a subset X of R , let S be the set of all elements in R that commute with every element in X. Then S is a subring of R , called the centralizer or commutant of X.

The center is the centralizer of the entire ring R. Elements or subsets of the center are said to be central in R ; they generate a subring of the center. The definition of an ideal in a ring is analogous to that of normal subgroup in a group. But, in actuality, it plays a role of an idealized generalization of an element in a ring; hence, the name "ideal". Like elements of rings, the study of ideals is central to structural understanding of a ring.

Let R be a ring. A subset I is said to be a two-sided ideal or simply ideal if it is both a left ideal and right ideal.

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A one-sided or two-sided ideal is then an additive subgroup of R. Similarly, one can consider the right ideal or the two-sided ideal generated by a subset of R. For example, the set of all positive and negative multiples of 2 along with 0 form an ideal of the integers, and this ideal is generated by the integer 2. In fact, every ideal of the ring of integers is principal. Like a group, a ring is said to be simple if it is nonzero and it has no proper nonzero two-sided ideals. A commutative simple ring is precisely a field.

Rings are often studied with special conditions set upon their ideals. For example, a ring in which there is no strictly increasing infinite chain of left ideals is called a left Noetherian ring. A ring in which there is no strictly decreasing infinite chain of left ideals is called a left Artinian ring. It is a somewhat surprising fact that a left Artinian ring is left Noetherian the Hopkins—Levitzki theorem. The integers, however, form a Noetherian ring which is not Artinian. For commutative rings, the ideals generalize the classical notion of divisibility and decomposition of an integer into prime numbers in algebra.

A ring homomorphism is said to be an isomorphism if there exists an inverse homomorphism to f i. Any bijective ring homomorphism is a ring isomorphism. A ring homomorphism between the same ring is called an endomorphism and an isomorphism between the same ring an automorphism. The kernel is a two-sided ideal of R. The image of f , on the other hand, is not always an ideal, but it is always a subring of S.

To give a ring homomorphism from a commutative ring R to a ring A with image contained in the center of A is the same as to give a structure of an algebra over R to A in particular gives a structure of A -module. The quotient ring of a ring, is analogous to the notion of a quotient group of a group. The last fact implies that actually any surjective ring homomorphism satisfies the universal property since the image of such a map is a quotient ring.

The concept of a module over a ring generalizes the concept of a vector space over a field by generalizing from multiplication of vectors with elements of a field scalar multiplication to multiplication with elements of a ring. This operation is commonly denoted multiplicatively and called multiplication. The axioms of modules are the following: for all a , b in R and all x , y in M , we have:. When the ring is noncommutative these axioms define left modules ; right modules are defined similarly by writing xa instead of ax.

Although similarly defined, the theory of modules is much more complicated than that of vector space, mainly, because, unlike vector spaces, modules are not characterized up to an isomorphism by a single invariant the dimension of a vector space. In particular, not all modules have a basis. Using this and denoting repeated addition by a multiplication by a positive integer allows identifying abelian groups with modules over the ring of integers.

In particular, every ring is an algebra over the integers. Let R and S be rings. Then the Chinese remainder theorem says there is a canonical ring isomorphism:. A "finite" direct product may also be viewed as a direct sum of ideals. Clearly the direct sum of such ideals also defines a product of rings that is isomorphic to R.

Equivalently, the above can be done through central idempotents. Assume R has the above decomposition. Then we can write. Again, one can reverse the construction. An important application of an infinite direct product is the construction of a projective limit of rings see below. Another application is a restricted product of a family of rings cf. Given a symbol t called a variable and a commutative ring R , the set of polynomials. It is called the polynomial ring over R. Given an element x of S , one can consider the ring homomorphism. Example: let f be a polynomial in one variable; i.

The substitution is a special case of the universal property of a polynomial ring.

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To give an example, let S be the ring of all functions from R to itself; the addition and the multiplication are those of functions. Let x be the identity function. The universal property says that this map extends uniquely to. The resulting map is injective if and only if R is infinite. Let k be an algebraically closed field. In particular, many local problems in algebraic geometry may be attacked through the study of the generators of an ideal in a polynomial ring.

There are some other related constructions. A formal power series ring does not have the universal property of a polynomial ring; a series may not converge after a substitution. The important advantage of a formal power series ring over a polynomial ring is that it is local in fact, complete. Let R be a ring not necessarily commutative. The set of all square matrices of size n with entries in R forms a ring with the entry-wise addition and the usual matrix multiplication.

It is called the matrix ring and is denoted by M n R. The Artin—Wedderburn theorem states any semisimple ring cf. A ring R and the matrix ring M n R over it are Morita equivalent : the category of right modules of R is equivalent to the category of right modules over M n R. Any commutative ring is the colimit of finitely generated subrings. A projective limit or a filtered limit of rings is defined as follows. The localization generalizes the construction of the field of fractions of an integral domain to an arbitrary ring and modules. The localization is frequently applied to a commutative ring R with respect to the complement of a prime ideal or a union of prime ideals in R.

This is the reason for the terminology "localization". The field of fractions of an integral domain R is the localization of R at the prime ideal zero. The most important properties of localization are the following: when R is a commutative ring and S a multiplicatively closed subset. In category theory, a localization of a category amounts to making some morphisms isomorphisms.

An element in a commutative ring R may be thought of as an endomorphism of any R -module. Thus, categorically, a localization of R with respect to a subset S of R is a functor from the category of R -modules to itself that sends elements of S viewed as endomorphisms to automorphisms and is universal with respect to this property. Let R be a commutative ring, and let I be an ideal of R.

The latter homomorphism is injective if R is a noetherian integral domain and I is a proper ideal, or if R is a noetherian local ring with maximal ideal I , by Krull's intersection theorem. The basic example is the completion Z p of Z at the principal ideal p generated by a prime number p ; it is called the ring of p -adic integers. The completion can in this case be constructed also from the p -adic absolute value on Q. It defines a distance function on Q and the completion of Q as a metric space is denoted by Q p.

It is again a field since the field operations extend to the completion. A complete ring has much simpler structure than a commutative ring. This owns to the Cohen structure theorem , which says, roughly, that a complete local ring tends to look like a formal power series ring or a quotient of it. On the other hand, the interaction between the integral closure and completion has been among the most important aspects that distinguish modern commutative ring theory from the classical one developed by the likes of Noether.

Pathological examples found by Nagata led to the reexamination of the roles of Noetherian rings and motivated, among other things, the definition of excellent ring. The most general way to construct a ring is by specifying generators and relations. Let F be a free ring i. Just as in the group case, every ring can be represented as a quotient of a free ring. Now, we can impose relations among symbols in X by taking a quotient. Explicitly, if E is a subset of F , then the quotient ring of F by the ideal generated by E is called the ring with generators X and relations E.

If we used a ring, say, A as a base ring instead of Z , then the resulting ring will be over A. Let A , B be algebras over a commutative ring R. See also: tensor product of algebras , change of rings. A nonzero ring with no nonzero zero-divisors is called a domain. A commutative domain is called an integral domain. The most important integral domains are principal ideals domains, PID for short, and fields. A principal ideal domain is an integral domain in which every ideal is principal.

An important class of integral domains that contain a PID is a unique factorization domain UFD , an integral domain in which every nonunit element is a product of prime elements an element is prime if it generates a prime ideal. The fundamental question in algebraic number theory is on the extent to which the ring of generalized integers in a number field , where an "ideal" admits prime factorization, fails to be a PID. Among theorems concerning a PID, the most important one is the structure theorem for finitely generated modules over a principal ideal domain.

The theorem may be illustrated by the following application to linear algebra. In algebraic geometry, UFDs arise because of smoothness. More precisely, a point in a variety over a perfect field is smooth if the local ring at the point is a regular local ring. A regular local ring is a UFD. The following is a chain of class inclusions that describes the relationship between rings, domains and fields:. A division ring is a ring such that every non-zero element is a unit. A commutative division ring is a field. A prominent example of a division ring that is not a field is the ring of quaternions.

Any centralizer in a division ring is also a division ring. In particular, the center of a division ring is a field. It turned out that every finite domain in particular finite division ring is a field; in particular commutative the Wedderburn's little theorem. Every module over a division ring is a free module has a basis ; consequently, much of linear algebra can be carried out over a division ring instead of a field. The study of conjugacy classes figures prominently in the classical theory of division rings. Cartan famously asked the following question: given a division ring D and a proper sub-division-ring S that is not contained in the center, does each inner automorphism of D restrict to an automorphism of S?

The answer is negative: this is the Cartan—Brauer—Hua theorem. A cyclic algebra , introduced by L. Dickson , is a generalization of a quaternion algebra. A ring is called a semisimple ring if it is semisimple as a left module or right module over itself; i. A ring is called a semiprimitive ring if its Jacobson radical is zero.

The Jacobson radical is the intersection of all maximal left ideals. A ring is semisimple if and only if it is artinian and is semiprimitive. An algebra over a field k is artinian if and only if it has finite dimension. Thus, a semisimple algebra over a field is necessarily finite-dimensional, while a simple algebra may have infinite dimension; e.

Any module over a semisimple ring is semisimple. Proof: any free module over a semisimple ring is clearly semisimple and any module is a quotient of a free module. Semisimplicity is closely related to separability.

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If A happens to be a field, then this is equivalent to the usual definition in field theory cf. For a field k , a k -algebra is central if its center is k and is simple if it is a simple ring. Since the center of a simple k -algebra is a field, any simple k -algebra is a central simple algebra over its center. In this section, a central simple algebra is assumed to have finite dimension. Also, we mostly fix the base field; thus, an algebra refers to a k -algebra.

The Skolem—Noether theorem states any automorphism of a central simple algebra is inner. By the Artin—Wedderburn theorem , a central simple algebra is the matrix ring of a division ring; thus, each similarity class is represented by a unique division ring. Tsen's theorem.

Finally, if k is a nonarchimedean local field e. Azumaya algebras generalize the notion of central simple algebras to a commutative local ring. See also: Novikov ring and uniserial ring.

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A ring may be viewed as an abelian group by using the addition operation , with extra structure: namely, ring multiplication. In the same way, there are other mathematical objects which may be considered as rings with extra structure. For example:. Many different kinds of mathematical objects can be fruitfully analyzed in terms of some associated ring.

To any topological space X one can associate its integral cohomology ring. Cohomology groups were later defined in terms of homology groups in a way which is roughly analogous to the dual of a vector space. To know each individual integral homology group is essentially the same as knowing each individual integral cohomology group, because of the universal coefficient theorem. The ring structure in cohomology provides the foundation for characteristic classes of fiber bundles , intersection theory on manifolds and algebraic varieties , Schubert calculus and much more.

To any group is associated its Burnside ring which uses a ring to describe the various ways the group can act on a finite set. 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 R 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 a in a domain is called irreducible if the only way of expressing it as a product. An example, important in field theory , are irreducible polynomials , i. The fact that Z 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 a is a prime element if whenever a divides a product bc , a divides b or c.

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In a domain, being prime implies being irreducible. The converse is true in a unique factorization domain, but false in general. It is the basis of modular arithmetic. 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. Except for the zero ring , any ring with identity possesses at least one maximal ideal; this follows from Zorn's lemma.

A ring is called Noetherian in honor of Emmy Noether , who developed this concept if every ascending chain of ideals. Equivalently, any ideal is generated by finitely many elements, or, yet equivalent, submodules of finitely generated modules are finitely generated. Being Noetherian is a highly important finiteness condition, and the condition is preserved under many operations that occur frequently in geometry. Any non-noetherian ring R is the union of its Noetherian subrings.

This fact, known as Noetherian approximation , allows the extension of certain theorems to non-Noetherian rings. A ring is called Artinian after Emil Artin , if every descending chain of ideals. 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.

In fact, by the Hopkins—Levitzki theorem , every Artinian ring is Noetherian. More precisely, Artinian rings can be characterized as the Noetherian rings whose Krull dimension is zero. As was mentioned above, Z is a unique factorization domain. This is not true for more general rings, as algebraists realized in the 19th century. For example, in. Prime ideals, as opposed to prime elements, provide a way to circumvent this problem.


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A prime ideal is a proper i. The opposite conclusion holds for any ideal, by definition. Thus, if a prime ideal is principal, it is equivalently generated by a prime element. This limits the usage of prime elements in ring theory. Any maximal ideal is a prime ideal or, more briefly, is prime. Proving that an ideal is prime, or equivalently that a ring has no zero-divisors can be very difficult. This ring has only one maximal ideal, namely pR p. Such rings are called local.

The spectrum of a ring R , [nb 1] denoted by Spec R , is the set of all prime ideals of R. It is equipped with a topology, the Zariski topology , which reflects the algebraic properties of R : a basis of open subsets is given by. Interpreting f as a function that takes the value f mod p i. The spectrum contains the set of maximal ideals, which is occasionally denoted mSpec R.

For an algebraically closed field k , mSpec k[ T 1 , 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. 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. The datum of the space and the sheaf is called an affine scheme. 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 R n , 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. 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. 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. The kernel is an ideal of R , and the image is a subring of S. 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. By means of this map, an integer n can be regarded as an element of R. For example, the binomial formula. Given two R -algebras S and T , their tensor product. 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 s 1 , 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 s 1 , 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. 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. Informally, the elements of m can be thought of as functions which vanish at the point p , whereas m 2 contains the ones which vanish with order at least 2. For any Noetherian local ring R , the inequality. 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.

Broadly speaking, regular local rings are somewhat similar to polynomial rings. 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. By Krull's principal ideal theorem , a foundational result in the dimension theory of rings , the dimension of.

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. As of , it is in general unknown, whether curves in three-dimensional space are set-theoretic complete intersections. The depth of a local ring R is the number of elements in some or, as can be shown, any maximal regular sequence, i. 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.

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.