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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. 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.


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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.

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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.

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.


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  • 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.

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    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.

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    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.

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    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.

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    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. 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. 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. The Burnside ring's additive group is the free abelian group whose basis are the transitive actions of the group and whose addition is the disjoint union of the action.

    Expressing an action in terms of the basis is decomposing an action into its transitive constituents. The multiplication is easily expressed in terms of the representation ring : the multiplication in the Burnside ring is formed by writing the tensor product of two permutation modules as a permutation module. The ring structure allows a formal way of subtracting one action from another.

    Since the Burnside ring is contained as a finite index subring of the representation ring, one can pass easily from one to the other by extending the coefficients from integers to the rational numbers. To any group ring or Hopf algebra is associated its representation ring or "Green ring". The representation ring's additive group is the free abelian group whose basis are the indecomposable modules and whose addition corresponds to the direct sum. Expressing a module in terms of the basis is finding an indecomposable decomposition of the module. The multiplication is the tensor product.

    When the algebra is semisimple, the representation ring is just the character ring from character theory , which is more or less the Grothendieck group given a ring structure. To any irreducible algebraic variety is associated its function field. The points of an algebraic variety correspond to valuation rings contained in the function field and containing the coordinate ring. The study of algebraic geometry makes heavy use of commutative algebra to study geometric concepts in terms of ring-theoretic properties. Birational geometry studies maps between the subrings of the function field.

    Every simplicial complex has an associated face ring, also called its Stanley—Reisner ring. This ring reflects many of the combinatorial properties of the simplicial complex, so it is of particular interest in algebraic combinatorics. In particular, the algebraic geometry of the Stanley—Reisner ring was used to characterize the numbers of faces in each dimension of simplicial polytopes. The monoid action of a ring R on an abelian group is simply an R -module. Essentially, an R -module is a generalization of the notion of a vector space — where rather than a vector space over a field, one has a "vector space over a ring".

    Therefore, associated to any abelian group, is a ring. Consider those endomorphisms of A , that "factor through" right or left multiplication of R. It was seen that every r in R gives rise to a morphism of A : right multiplication by r. It is in fact true that this association of any element of R , to a morphism of A , as a function from R to End R A , is an isomorphism of rings.

    In this sense, therefore, any ring can be viewed as the endomorphism ring of some abelian X -group by X -group, it is meant a group with X being its set of operators. Any ring can be seen as a preadditive category with a single object. It is therefore natural to consider arbitrary preadditive categories to be generalizations of rings. And indeed, many definitions and theorems originally given for rings can be translated to this more general context. Additive functors between preadditive categories generalize the concept of ring homomorphism, and ideals in additive categories can be defined as sets of morphisms closed under addition and under composition with arbitrary morphisms.

    Algebraists have defined structures more general than rings by weakening or dropping some of ring axioms. A rng is the same as a ring, except that the existence of a multiplicative identity is not assumed. A nonassociative ring is an algebraic structure that satisfies all of the ring axioms except the associative property and the existence of a multiplicative identity. A notable example is a Lie algebra. There exists some structure theory for such algebras that generalizes the analogous results for Lie algebras and associative algebras.

    Let C be a category with finite products. Let pt denote a terminal object of C an empty product. Euler's Phi-Function IV. Groups 14 Elementary Properties 15 Generators. Direct Products 16 Cosets 17 Lagrange's Theorem. Group Homomorphisms 21 Homomorphisms of Groups. Subrings 26 Fields 27 Isomorphism. Characteristic VII. Quotient Rings 38 Homomorphisms of Rings. Galois Theory: Overview 42 Simple Extensions.

    Solvable and Alternating Groups 54 Isomorphis Theorems. Sets B. Proofs C. Mathematical Induction D. Linear Algebra E. Du kanske gillar. Lifespan David Sinclair Inbunden. Inbunden Engelska, Spara som favorit. Skickas inom vardagar. Engineers and computer scientists who need a basic understanding of algebra will benefit from this accessible book.