Direct product Let and be rings. Then the
product can be equipped with the following natural ring structure: : \begin{align} & (r_1,s_1) + (r_2,s_2) = (r_1+r_2,s_1+s_2) \\ & (r_1,s_1) \cdot (r_2,s_2)=(r_1\cdot r_2,s_1\cdot s_2) \end{align} for all in and in . The ring with the above operations of addition and multiplication and the multiplicative identity is called the
direct product of with . The same construction also works for an arbitrary family of rings: if are rings indexed by a set , then \prod_{i \in I} R_i is a ring with componentwise addition and multiplication. Let be a commutative ring and \mathfrak{a}_1, \cdots, \mathfrak{a}_n be ideals such that \mathfrak{a}_i + \mathfrak{a}_j = (1) whenever . Then the
Chinese remainder theorem says there is a canonical ring isomorphism: R /{\textstyle \bigcap_{i=1}^{n}{\mathfrak{a}_i}} \simeq \prod_{i=1}^{n}{R/ \mathfrak{a}_i}, \qquad x \bmod {\textstyle \bigcap_{i=1}^{n}\mathfrak{a}_i} \mapsto (x \bmod \mathfrak{a}_1, \ldots , x \bmod \mathfrak{a}_n). A "finite" direct product may also be viewed as a direct sum of ideals. Namely, let R_i, 1 \le i \le n be rings, R_i \to R = \prod R_i the inclusions with the images \mathfrak{a}_i (in particular \mathfrak{a}_i are rings though not subrings). Then \mathfrak{a}_i are ideals of and R = \mathfrak{a}_1 \oplus \cdots \oplus \mathfrak{a}_n, \quad \mathfrak{a}_i \mathfrak{a}_j = 0, i \ne j, \quad \mathfrak{a}_i^2 \subseteq \mathfrak{a}_i as a direct sum of abelian groups (because for abelian groups finite products are the same as direct sums). Clearly the direct sum of such ideals also defines a product of rings that is isomorphic to . Equivalently, the above can be done through
central idempotents. Assume that has the above decomposition. Then we can write 1 = e_1 + \cdots + e_n, \quad e_i \in \mathfrak{a}_i. By the conditions on \mathfrak{a}_i, one has that are central idempotents and , (orthogonal). Again, one can reverse the construction. Namely, if one is given a partition of 1 in orthogonal central idempotents, then let \mathfrak{a}_i = R e_i, which are two-sided ideals. If each is not a sum of orthogonal central idempotents, then their direct sum is isomorphic to . 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.
adele ring).
Polynomial ring Given a symbol (called a variable) and a commutative ring , the set of polynomials : R[t] = \left\{ a_n t^n + a_{n-1} t^{n -1} + \dots + a_1 t + a_0 \mid n \ge 0, a_j \in R \right\} forms a commutative ring with the usual addition and multiplication, containing as a subring. It is called the
polynomial ring over . More generally, the set R\left[t_1, \ldots, t_n\right] of all polynomials in variables t_1, \ldots, t_n forms a commutative ring, containing R\left[t_i\right] as subrings. If is an
integral domain, then is also an integral domain; its field of fractions is the field of
rational functions. If is a Noetherian ring, then is a Noetherian ring. If is a unique factorization domain, then is a unique factorization domain. Finally, is a field if and only if is a principal ideal domain. Let R \subseteq S be commutative rings. Given an element of , one can consider the ring homomorphism : R[t] \to S, \quad f \mapsto f(x) (that is, the
substitution). If and , then . Because of this, the polynomial is often also denoted by . The image of the map is denoted by ; it is the same thing as the subring of generated by and . Example: k\left[t^2, t^3\right] denotes the image of the homomorphism :k[x, y] \to k[t], \, f \mapsto f\left(t^2, t^3\right). In other words, it is the subalgebra of generated by and . Example: let be a polynomial in one variable, that is, an element in a polynomial ring . Then is an element in and is divisible by in that ring. The result of substituting zero to in is , the derivative of at . The substitution is a special case of the universal property of a polynomial ring. The property states: given a ring homomorphism \phi: R \to S and an element in there exists a unique ring homomorphism \overline{\phi}: R[t] \to S such that \overline{\phi}(t) = x and \overline{\phi} restricts to . For example, choosing a basis, a
symmetric algebra satisfies the universal property and so is a polynomial ring. To give an example, let be the ring of all functions from to itself; the addition and the multiplication are those of functions. Let be the identity function. Each in defines a constant function, giving rise to the homomorphism . The universal property says that this map extends uniquely to :R[t] \to S, \quad f \mapsto \overline{f} ( maps to ) where \overline{f} is the
polynomial function defined by . The resulting map is injective if and only if is infinite. Given a non-constant monic polynomial in , there exists a ring containing such that is a product of linear factors in . Let be an algebraically closed field. The
Hilbert's Nullstellensatz (theorem of zeros) states that there is a natural one-to-one correspondence between the set of all prime ideals in k\left[t_1, \ldots, t_n\right] and the set of closed subvarieties of . In particular, many local problems in algebraic geometry may be attacked through the study of the generators of an ideal in a polynomial ring. (cf.
Gröbner basis.) There are some other related constructions. A
formal power series ring R[\![t]\!] consists of formal power series : \sum_0^\infty a_i t^i, \quad a_i \in R together with multiplication and addition that mimic those for convergent series. It contains as a subring. 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).
Matrix ring and endomorphism ring Let be a ring (not necessarily commutative). The set of all square matrices of size with entries in forms a ring with the entry-wise addition and the usual matrix multiplication. It is called the
matrix ring and is denoted by . Given a right -module , the set of all -linear maps from to itself forms a ring with addition that is of function and multiplication that is of
composition of functions; it is called the endomorphism ring of and is denoted by . As in linear algebra, a matrix ring may be canonically interpreted as an endomorphism ring: \operatorname{End}_R(R^n) \simeq \operatorname{M}_n(R). This is a special case of the following fact: If f: \oplus_1^n U \to \oplus_1^n U is an -linear map, then may be written as a matrix with entries in , resulting in the ring isomorphism: :\operatorname{End}_R(\oplus_1^n U) \to \operatorname{M}_n(S), \quad f \mapsto (f_{ij}). Any ring homomorphism induces .
Schur's lemma says that if is a simple right -module, then is a division ring. If U = \bigoplus_{i = 1}^r U_i^{\oplus m_i} is a direct sum of -copies of simple -modules U_i, then :\operatorname{End}_R(U) \simeq \prod_{i=1}^r \operatorname{M}_{m_i} (\operatorname{End}_R(U_i)). The
Artin–Wedderburn theorem states any
semisimple ring (cf. below) is of this form. A ring and the matrix ring over it are
Morita equivalent: the
category of right modules of is equivalent to the category of right modules over . In particular, two-sided ideals in correspond in one-to-one to two-sided ideals in .
Limits and colimits of rings Let be a sequence of rings such that is a subring of for all . Then the union (or
filtered colimit) of is the ring \varinjlim R_i defined as follows: it is the disjoint union of all 's modulo the equivalence relation if and only if in for sufficiently large . Examples of colimits: • A polynomial ring in infinitely many variables: R[t_1, t_2, \cdots] = \varinjlim R[t_1, t_2, \cdots, t_m]. • The
algebraic closure of
finite fields of the same characteristic \overline{\mathbf{F}}_p = \varinjlim \mathbf{F}_{p^m}. • The
function field of an algebraic variety over a field is \varinjlim k[U] where the limit runs over all the coordinate rings of nonempty open subsets (more succinctly it is the
stalk of the structure sheaf at the
generic point.) Any ring is the filtered colimit (union) of its
finitely generated subrings. A
projective limit (or a
filtered limit) of rings is defined as follows. Suppose we are given a family of rings , running over positive integers, say, and ring homomorphisms , such that are all the identities and is whenever . Then \varprojlim R_i is the subring of \textstyle \prod R_i consisting of such that maps to under , . For an example of a projective limit, see ''''.
Localization The
localization generalizes the construction of the
field of fractions of an integral domain to an arbitrary ring and modules. Given a (not necessarily commutative) ring and a subset of , there exists a ring R[S^{-1}] together with the ring homomorphism R \to R\left[S^{-1}\right] that "inverts" ; that is, the homomorphism maps elements in to unit elements in R\left[S^{-1}\right], and, moreover, any ring homomorphism from that "inverts" uniquely factors through R\left[S^{-1}\right]. The ring R\left[S^{-1}\right] is called the
localization of with respect to . For example, if is a commutative ring and an element in , then the localization R\left[f^{-1}\right] consists of elements of the form r/f^n, \, r \in R , \, n \ge 0 (to be precise, R\left[f^{-1}\right] = R[t]/(tf - 1).) The localization is frequently applied to a commutative ring with respect to the complement of a prime ideal (or a union of prime ideals) in . In that case S = R - \mathfrak{p}, one often writes R_\mathfrak{p} for R\left[S^{-1}\right]. R_\mathfrak{p} is then a
local ring with the
maximal ideal \mathfrak{p} R_\mathfrak{p}. This is the reason for the terminology "localization". The field of fractions of an integral domain is the localization of at the prime ideal zero. If \mathfrak{p} is a prime ideal of a commutative ring , then the field of fractions of R/\mathfrak{p} is the same as the residue field of the local ring R_\mathfrak{p} and is denoted by k(\mathfrak{p}). If is a left -module, then the localization of with respect to is given by a
change of rings M\left[S^{-1}\right] = R\left[S^{-1}\right] \otimes_R M.M \to M\left[S^{-1}\right], \, m \mapsto m / 1. Its kernel consists of elements such that for some in . --> The most important properties of localization are the following: when is a commutative ring and a multiplicatively closed subset • \mathfrak{p} \mapsto \mathfrak{p}\left[S^{-1}\right] is a bijection between the set of all prime ideals in disjoint from and the set of all prime ideals in R\left[S^{-1}\right]. • R\left[S^{-1}\right] = \varinjlim R\left[f^{-1}\right], running over elements in with partial ordering given by divisibility. • The localization is exact: 0 \to M'\left[S^{-1}\right] \to M\left[S^{-1}\right] \to M''\left[S^{-1}\right] \to 0 is exact over R\left[S^{-1}\right] whenever 0 \to M' \to M \to M'' \to 0 is exact over . • Conversely, if 0 \to M'_\mathfrak{m} \to M_\mathfrak{m} \to M''_\mathfrak{m} \to 0 is exact for any maximal ideal \mathfrak{m}, then 0 \to M' \to M \to M'' \to 0 is exact. • A remark: localization is no help in proving a global existence. One instance of this is that if two modules are isomorphic at all prime ideals, it does not follow that they are isomorphic. (One way to explain this is that the localization allows one to view a module as a sheaf over prime ideals and a sheaf is inherently a local notion.) In
category theory, a
localization of a category amounts to making some morphisms isomorphisms. An element in a commutative ring may be thought of as an endomorphism of any -module. Thus, categorically, a localization of with respect to a subset of is a
functor from the category of -modules to itself that sends elements of viewed as endomorphisms to automorphisms and is universal with respect to this property. (Of course, then maps to R\left[S^{-1}\right] and -modules map to R\left[S^{-1}\right]-modules.)
Completion Let be a commutative ring, and let be an ideal of . The
completion of at is the projective limit \hat{R} = \varprojlim R/I^n; it is a commutative ring. The canonical homomorphisms from to the quotients R/I^n induce a homomorphism R \to \hat{R}. The latter homomorphism is injective if is a Noetherian integral domain and is a proper ideal, or if is a Noetherian local ring with maximal ideal , by
Krull's intersection theorem. The construction is especially useful when is a maximal ideal. The basic example is the completion of at the principal ideal generated by a prime number ; it is called the ring of
-adic integers and is denoted The completion can in this case be constructed also from the
-adic absolute value on The -adic absolute value on is a map x \mapsto |x| from to given by |n|_p=p^{-v_p(n)} where v_p(n) denotes the exponent of in the prime factorization of a nonzero integer into prime numbers (we also put |0|_p=0 and |m/n|_p = |m|_p/|n|_p). It defines a distance function on and the completion of as a
metric space is denoted by It is again a field since the field operations extend to the completion. The subring of consisting of elements with is isomorphic to Similarly, the formal power series ring {{math|
R[{[
t]}]}} is the completion of at (see also ''
Hensel's lemma'') 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.
Rings with generators and relations The most general way to construct a ring is by specifying generators and relations. Let be a
free ring (that is, free algebra over the integers) with the set of symbols, that is, consists of polynomials with integral coefficients in noncommuting variables that are elements of . A free ring satisfies the universal property: any function from the set to a ring factors through so that is the unique ring homomorphism. 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 by taking a quotient. Explicitly, if is a subset of , then the quotient ring of by the ideal generated by is called the ring with generators and relations . If we used a ring, say, as a base ring instead of then the resulting ring will be over . For example, if E = \{ xy - yx \mid x, y \in X \}, then the resulting ring will be the usual polynomial ring with coefficients in in variables that are elements of (It is also the same thing as the
symmetric algebra over with symbols .) In the category-theoretic terms, the formation S \mapsto \text{the free ring generated by the set } S is the left adjoint functor of the
forgetful functor from the
category of rings to
Set (and it is often called the free ring functor.) Let , be algebras over a commutative ring . Then the tensor product of -modules A \otimes_R B is an -algebra with multiplication characterized by (x \otimes u) (y \otimes v) = xy \otimes uv. == Special kinds of rings ==