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Given a subset E of R , the smallest subring of R containing E is the intersection of all subrings of R containing E , and it is called the subring generated by E.
For a ring R , the smallest subring of R 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 each individually 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.
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 that is, a ring homomorphism that is an inverse function.
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 which in particular gives a structure of an 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, that is, an element in a polynomial ring R.
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. 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.
Gröbner basis. 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 that is, free algebra over the integers with the set X of symbols, that is, F consists of polynomials with integral coefficients in noncommuting variables that are elements of X.
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.
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