Editing Ring (mathematics) (section)

== Constructions ==

=== Direct product ===
{{Main|Direct product of rings}}

Let {{mvar|R}} and {{mvar|S}} be rings. Then the [[cartesian product|product]] {{math|''R'' × ''S''}} can be equipped with the following natural ring structure:
: <math>\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}</math>
for all {{math|''r''{{sub|1}}, ''r''{{sub|2}}}} in {{mvar|R}} and {{math|''s''{{sub|1}}, ''s''{{sub|2}}}} in&nbsp;{{mvar|S}}. The ring {{math|''R'' × ''S''}} with the above operations of addition and multiplication and the multiplicative identity {{math|(1, 1)}} is called the '''[[Direct product of rings|direct product]]''' of {{mvar|R}} with&nbsp;{{mvar|S}}. The same construction also works for an arbitrary family of rings: if {{mvar|R{{sub|i}}}} are rings indexed by a set {{mvar|I}}, then <math display="inline"> \prod_{i \in I} R_i</math> is a ring with componentwise addition and multiplication.

Let {{mvar|R}} be a commutative ring and <math>\mathfrak{a}_1, \cdots, \mathfrak{a}_n</math> be ideals such that <math>\mathfrak{a}_i + \mathfrak{a}_j = (1)</math> whenever {{math|''i'' ≠ ''j''}}. Then the [[Chinese remainder theorem]] says there is a canonical ring isomorphism:
<math display="block">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).</math>

A "finite" direct product may also be viewed as a direct sum of ideals.{{sfnp|Cohn|2003|loc=Theorem 4.5.1|ps=}} Namely, let <math>R_i, 1 \le i \le n</math> be rings, <math display="inline">R_i \to R = \prod R_i</math> the inclusions with the images <math>\mathfrak{a}_i</math> (in particular <math>\mathfrak{a}_i</math> are rings though not subrings). Then <math>\mathfrak{a}_i</math> are ideals of {{mvar|R}} and
<math display="block">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</math>
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&nbsp;{{mvar|R}}. Equivalently, the above can be done through [[central idempotent]]s. Assume that {{mvar|R}} has the above decomposition. Then we can write
<math display="block">1 = e_1 + \cdots + e_n, \quad e_i \in \mathfrak{a}_i.</math>
By the conditions on <math>\mathfrak{a}_i,</math> one has that {{mvar|e{{sub|i}}}} are central idempotents and {{math|1=''e{{sub|i}}e{{sub|j}}'' = 0}}, {{math|''i'' ≠ ''j''}} (orthogonal). Again, one can reverse the construction. Namely, if one is given a partition of 1 in orthogonal central idempotents, then let <math>\mathfrak{a}_i = R e_i,</math> which are two-sided ideals. If each {{mvar|e{{sub|i}}}} is not a sum of orthogonal central idempotents,{{efn|Such a central idempotent is called [[centrally primitive]].}} then their direct sum is isomorphic to&nbsp;{{mvar|R}}.

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 ===
{{main|Polynomial ring}}

Given a symbol {{mvar|t}} (called a variable) and a commutative ring&nbsp;{{mvar|R}}, the set of polynomials
: <math>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\}</math>
forms a commutative ring with the usual addition and multiplication, containing {{mvar|R}} as a subring. It is called the [[polynomial ring]] over&nbsp;{{mvar|R}}. More generally, the set <math>R\left[t_1, \ldots, t_n\right]</math> of all polynomials in variables <math>t_1, \ldots, t_n</math> forms a commutative ring, containing <math>R\left[t_i\right]</math> as subrings.

If {{mvar|R}} is an [[integral domain]], then {{math|''R''[''t'']}} is also an integral domain; its field of fractions is the field of [[rational function]]s. If {{mvar|R}} is a Noetherian ring, then {{math|''R''[''t'']}} is a Noetherian ring. If {{mvar|R}} is a unique factorization domain, then {{math|''R''[''t'']}} is a unique factorization domain. Finally, {{mvar|R}} is a field if and only if {{math|''R''[''t'']}} is a principal ideal domain.

Let <math>R \subseteq S</math> be commutative rings. Given an element {{mvar|x}} of&nbsp;{{mvar|S}}, one can consider the ring homomorphism
: <math>R[t] \to S, \quad f \mapsto f(x)</math>

(that is, the [[substitution (algebra)|substitution]]). If {{math|1=''S'' = ''R''[''t'']}} and {{math|1=''x'' = ''t''}}, then {{math|1=''f''(''t'') = ''f''}}. Because of this, the polynomial {{mvar|f}} is often also denoted by {{math|''f''(''t'')}}. The image of the map {{tmath|f \mapsto f(x)}} is denoted by {{math|''R''[''x'']}}; it is the same thing as the subring of {{mvar|S}} generated by {{mvar|R}} and&nbsp;{{mvar|x}}.

Example: <math>k\left[t^2, t^3\right]</math> denotes the image of the homomorphism
:<math>k[x, y] \to k[t], \, f \mapsto f\left(t^2, t^3\right).</math>

In other words, it is the subalgebra of {{math|''k''[''t'']}} generated by {{math|''t''{{sup|2}}}} and&nbsp;{{math|''t''{{sup|3}}}}.

Example: let {{mvar|f}} be a polynomial in one variable, that is, an element in a polynomial ring {{mvar|R}}. Then {{math|''f''(''x'' + ''h'')}} is an element in {{math|''R''[''h'']}} and {{math|''f''(''x'' + ''h'') – ''f''(''x'')}} is divisible by {{mvar|h}} in that ring. The result of substituting zero to {{mvar|h}} in {{math|(''f''(''x'' + ''h'') – ''f''(''x'')) / ''h''}} is {{math|''f' ''(''x'')}}, the derivative of {{mvar|f}} at&nbsp;{{mvar|x}}.

The substitution is a special case of the universal property of a polynomial ring. The property states: given a ring homomorphism <math>\phi: R \to S</math> and an element {{mvar|x}} in {{mvar|S}} there exists a unique ring homomorphism <math>\overline{\phi}: R[t] \to S</math> such that <math>\overline{\phi}(t) = x</math> and <math>\overline{\phi}</math> restricts to {{mvar|ϕ}}.{{sfnp|Jacobson|2009|p=122|loc=Theorem 2.10|ps=}} For example, choosing a basis, a [[symmetric algebra]] satisfies the universal property and so is a polynomial ring.

To give an example, let {{mvar|S}} be the ring of all functions from {{mvar|R}} to itself; the addition and the multiplication are those of functions. Let {{mvar|x}} be the identity function. Each {{mvar|r}} in {{mvar|R}} defines a constant function, giving rise to the homomorphism {{math|''R'' → ''S''}}. The universal property says that this map extends uniquely to
:<math>R[t] \to S, \quad f \mapsto \overline{f}</math>
({{mvar|t}} maps to {{mvar|x}}) where <math>\overline{f}</math> is the [[polynomial function]] defined by {{mvar|f}}. The resulting map is injective if and only if {{mvar|R}} is infinite.

Given a non-constant monic polynomial {{mvar|f}} in {{math|''R''[''t'']}}, there exists a ring {{mvar|S}} containing {{mvar|R}} such that {{mvar|f}} is a product of linear factors in {{math|''S''[''t'']}}.{{sfnp|Bourbaki|1964|loc=Ch 5. §1, Lemma 2|ps=}}

Let {{mvar|k}} 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 <math>k\left[t_1, \ldots, t_n\right]</math> and the set of closed subvarieties of {{mvar|k{{sup|n}}}}. 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]] <math>R[\![t]\!]</math> consists of formal power series
: <math>\sum_0^\infty a_i t^i, \quad a_i \in R</math>

together with multiplication and addition that mimic those for convergent series. It contains {{math|''R''[''t'']}} 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 ring|local]] (in fact, [[complete ring|complete]]).

=== Matrix ring and endomorphism ring ===
{{Main|Matrix ring|Endomorphism ring}}

Let {{mvar|R}} be a ring (not necessarily commutative). The set of all square matrices of size {{mvar|n}} with entries in {{mvar|R}} forms a ring with the entry-wise addition and the usual matrix multiplication. It is called the [[matrix ring]] and is denoted by {{math|M{{sub|''n''}}(''R'')}}. Given a right {{mvar|R}}-module {{mvar|U}}, the set of all {{mvar|R}}-linear maps from {{mvar|U}} 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 {{mvar|U}} and is denoted by {{math|End{{sub|''R''}}(''U'')}}.

As in linear algebra, a matrix ring may be canonically interpreted as an endomorphism ring: <math>\operatorname{End}_R(R^n) \simeq \operatorname{M}_n(R).</math> This is a special case of the following fact: If <math>f: \oplus_1^n U \to \oplus_1^n U</math> is an {{mvar|R}}-linear map, then {{mvar|f}} may be written as a matrix with entries {{mvar|f{{sub|ij}}}} in {{math|1=''S'' = End{{sub|''R''}}(''U'')}}, resulting in the ring isomorphism:
:<math>\operatorname{End}_R(\oplus_1^n U) \to \operatorname{M}_n(S), \quad f \mapsto (f_{ij}).</math>

Any ring homomorphism {{math|''R'' → ''S''}} induces {{math|M{{sub|''n''}}(''R'') → M{{sub|''n''}}(''S'')}}.{{sfnp|Cohn|2003|loc=4.4|ps=}}

[[Schur's lemma]] says that if {{mvar|U}} is a simple right {{mvar|R}}-module, then {{math|End{{sub|''R''}}(''U'')}} is a division ring.{{sfnp|Lang|2002|loc=Ch. XVII. Proposition 1.1|ps=}} If <math>U = \bigoplus_{i = 1}^r U_i^{\oplus m_i}</math> is a direct sum of {{mvar|m{{sub|i}}}}-copies of simple {{mvar|R}}-modules <math>U_i,</math> then
:<math>\operatorname{End}_R(U) \simeq \prod_{i=1}^r \operatorname{M}_{m_i} (\operatorname{End}_R(U_i)).</math>
The [[Artin–Wedderburn theorem]] states any [[semisimple ring]] (cf. below) is of this form.

A ring {{mvar|R}} and the matrix ring {{math|M{{sub|''n''}}(''R'')}} over it are [[Morita equivalent]]: the [[Category (mathematics)|category]] of right modules of {{mvar|R}} is equivalent to the category of right modules over {{math|M{{sub|''n''}}(''R'')}}.{{sfnp|Cohn|2003|loc=4.4|ps=}} In particular, two-sided ideals in {{mvar|R}} correspond in one-to-one to two-sided ideals in {{math|M{{sub|''n''}}(''R'')}}.

=== Limits and colimits of rings ===
Let {{mvar|R{{sub|i}}}} be a sequence of rings such that {{mvar|R{{sub|i}}}} is a subring of {{math|''R''{{sub|''i'' + 1}}}} for all {{mvar|i}}. Then the union (or [[filtered colimit]]) of {{mvar|R{{sub|i}}}} is the ring <math>\varinjlim R_i</math> defined as follows: it is the disjoint union of all {{mvar|R{{sub|i}}}}'s modulo the equivalence relation {{math|''x'' ~ ''y''}} if and only if {{math|1=''x'' = ''y''}} in {{mvar|R{{sub|i}}}} for sufficiently large {{mvar|i}}.

Examples of colimits:
* A polynomial ring in infinitely many variables: <math>R[t_1, t_2, \cdots] = \varinjlim R[t_1, t_2, \cdots, t_m].</math>
* The [[algebraic closure]] of [[finite field]]s of the same characteristic <math>\overline{\mathbf{F}}_p = \varinjlim \mathbf{F}_{p^m}.</math>
* The field of [[formal Laurent series]] over a field {{mvar|k}}: <math>k(\!(t)\!) = \varinjlim t^{-m}k[\![t]\!]</math> (it is the field of fractions of the [[formal power series ring]] <math>k[\![t]\!].</math>)
* The [[function field of an algebraic variety]] over a field {{mvar|k}} is <math>\varinjlim k[U]</math> where the limit runs over all the coordinate rings {{math|''k''[''U'']}} of nonempty open subsets {{mvar|U}} (more succinctly it is the [[stalk (mathematics)|stalk]] of the structure sheaf at the [[generic point]].)

Any commutative ring is the colimit of [[finitely generated ring|finitely generated subrings]].<!-- non-commutative case? -->

A [[projective limit]] (or a [[filtered limit]]) of rings is defined as follows. Suppose we are given a family of rings {{math|''R''{{sub|''i''}}}}, {{math|''i''}} running over positive integers, say, and ring homomorphisms {{math|''R''{{sub|''j''}} → ''R''{{sub|''i''}}}}, {{math|''j'' ≥ ''i''}} such that {{math|''R''{{sub|''i''}} → ''R''{{sub|''i''}}}} are all the identities and {{math|''R''{{sub|''k''}} → ''R''{{sub|''j''}} → ''R''{{sub|''i''}}}} is {{math|''R''{{sub|''k''}} → ''R''{{sub|''i''}}}} whenever {{math|''k'' ≥ ''j'' ≥ ''i''}}. Then <math>\varprojlim R_i</math> is the subring of <math>\textstyle \prod R_i</math> consisting of {{math|(''x''{{sub|''n''}})}} such that {{math|''x''{{sub|''j''}}}} maps to {{math|''x''{{sub|''i''}}}} under {{math|''R''{{sub|''j''}} → ''R''{{sub|''i''}}}}, {{math|''j'' ≥ ''i''}}.

For an example of a projective limit, see ''{{slink||Completion}}''.

=== Localization ===
The [[localization of a ring|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 {{mvar|R}} and a subset {{mvar|S}} of {{mvar|R}}, there exists a ring <math>R[S^{-1}]</math> together with the ring homomorphism <math>R \to R\left[S^{-1}\right]</math> that "inverts" {{mvar|S}}; that is, the homomorphism maps elements in {{mvar|S}} to unit elements in <math>R\left[S^{-1}\right],</math> and, moreover, any ring homomorphism from {{mvar|R}} that "inverts" {{mvar|S}} uniquely factors through <math>R\left[S^{-1}\right].</math>{{sfnp|Cohn|1995|loc=Proposition 1.3.1|ps=}} The ring <math>R\left[S^{-1}\right]</math> is called the '''localization''' of {{mvar|R}} with respect to {{mvar|S}}. For example, if {{mvar|R}} is a commutative ring and {{mvar|f}} an element in {{mvar|R}}, then the localization <math>R\left[f^{-1}\right]</math> consists of elements of the form <math>r/f^n, \, r \in R , \, n \ge 0</math> (to be precise, <math>R\left[f^{-1}\right] = R[t]/(tf - 1).</math>){{sfnp|Eisenbud|1995|loc=Exercise 2.2|ps=}}

The localization is frequently applied to a commutative ring {{mvar|R}} with respect to the complement of a prime ideal (or a union of prime ideals) in&nbsp;{{mvar|R}}. In that case <math>S = R - \mathfrak{p},</math> one often writes <math>R_\mathfrak{p}</math> for <math>R\left[S^{-1}\right].</math> <math>R_\mathfrak{p}</math> is then a [[local ring]] with the [[maximal ideal]] <math>\mathfrak{p} R_\mathfrak{p}.</math> This is the reason for the terminology "localization". The field of fractions of an integral domain {{mvar|R}} is the localization of {{mvar|R}} at the prime ideal zero. If <math>\mathfrak{p}</math> is a prime ideal of a commutative ring&nbsp;{{mvar|R}}, then the field of fractions of <math>R/\mathfrak{p}</math> is the same as the residue field of the local ring <math>R_\mathfrak{p}</math> and is denoted by <math>k(\mathfrak{p}).</math><!-- In algebraic geometry, the field of fractions is the localization at the "[[generic point]]" -->

If {{mvar|M}} is a left {{mvar|R}}-module, then the localization of {{mvar|M}} with respect to {{mvar|S}} is given by a [[change of rings]] <math>M\left[S^{-1}\right] = R\left[S^{-1}\right] \otimes_R M.</math><!-- intuitively, {{mvar|M{{sub|p}}}} corresponds to the fiber over {{mvar|p}} in&nbsp;{{mvar|M}}. In particular, there is the canonical map <math>M \to M\left[S^{-1}\right], \, m \mapsto m / 1.</math> Its kernel consists of elements {{mvar|m}} such that {{mvar|1=''sm'' = 0}} for some {{mvar|s}} in {{mvar|S}}. -->

The most important properties of localization are the following: when {{mvar|R}} is a commutative ring and {{mvar|S}} a multiplicatively closed subset
* <math>\mathfrak{p} \mapsto \mathfrak{p}\left[S^{-1}\right]</math> is a bijection between the set of all prime ideals in {{mvar|R}} disjoint from {{mvar|S}} and the set of all prime ideals in <math>R\left[S^{-1}\right].</math>{{sfnp|Milne|2012|loc=Proposition 6.4|ps=}}
* <math>R\left[S^{-1}\right] = \varinjlim R\left[f^{-1}\right],</math> {{mvar|f}} running over elements in {{mvar|S}} with partial ordering given by divisibility.{{sfnp|Milne|2012|loc=end of Chapter 7|ps=}}
* The localization is exact: <math display="block">0 \to M'\left[S^{-1}\right] \to M\left[S^{-1}\right] \to M''\left[S^{-1}\right] \to 0</math> is exact over <math>R\left[S^{-1}\right]</math> whenever <math>0 \to M' \to M \to M'' \to 0</math> is exact over&nbsp;{{mvar|R}}.
* Conversely, if <math>0 \to M'_\mathfrak{m} \to M_\mathfrak{m} \to M''_\mathfrak{m} \to 0</math> is exact for any maximal ideal <math>\mathfrak{m},</math> then <math>0 \to M' \to M \to M'' \to 0</math> 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 {{mvar|R}} may be thought of as an endomorphism of any {{mvar|R}}-module. Thus, categorically, a localization of {{mvar|R}} with respect to a subset {{mvar|S}} of {{mvar|R}} is a [[functor]] from the category of {{mvar|R}}-modules to itself that sends elements of {{mvar|S}} viewed as endomorphisms to automorphisms and is universal with respect to this property. (Of course, {{mvar|R}} then maps to <math>R\left[S^{-1}\right]</math> and {{mvar|R}}-modules map to <math>R\left[S^{-1}\right]</math>-modules.)

=== Completion ===
Let {{mvar|R}} be a commutative ring, and let {{mvar|I}} be an ideal of&nbsp;{{mvar|R}}.
The '''[[Completion (ring theory)|completion]]''' of {{mvar|R}} at {{mvar|I}} is the projective limit <math>\hat{R} = \varprojlim R/I^n;</math> it is a commutative ring. The canonical homomorphisms from {{mvar|R}} to the quotients <math>R/I^n</math> induce a homomorphism <math>R \to \hat{R}.</math> The latter homomorphism is injective if {{mvar|R}} is a Noetherian integral domain and {{mvar|I}} is a proper ideal, or if {{mvar|R}} is a Noetherian local ring with maximal ideal {{mvar|I}}, by [[Krull's intersection theorem]].{{sfnp|Atiyah|Macdonald|1969|loc=Theorem 10.17 and its corollaries|ps=}} The construction is especially useful when {{mvar|I}} is a maximal ideal.

The basic example is the completion of {{tmath|\Z}} at the principal ideal {{math|(''p'')}} generated by a prime number {{mvar|p}}; it is called the ring of [[p-adic integer|{{mvar|p}}-adic integers]] and is denoted {{tmath|\Z_p.}} The completion can in this case be constructed also from the [[p-adic absolute value|{{mvar|p}}-adic absolute value]] on {{tmath|\Q.}} The {{mvar|p}}-adic absolute value on {{tmath|\Q}} is a map <math>x \mapsto |x|</math> from {{tmath|\Q}} to {{tmath|\R}} given by <math>|n|_p=p^{-v_p(n)}</math> where <math>v_p(n)</math> denotes the exponent of {{mvar|p}} in the prime factorization of a nonzero integer {{mvar|n}} into prime numbers (we also put <math>|0|_p=0</math> and <math>|m/n|_p = |m|_p/|n|_p</math>). It defines a distance function on {{tmath|\Q}} and the completion of {{tmath|\Q}} as a [[metric space]] is denoted by {{tmath|\Q_p.}} It is again a field since the field operations extend to the completion. The subring of {{tmath|\Q_p}} consisting of elements {{mvar|x}} with {{math|{{abs|''x''}}{{sub|''p''}} ≤ 1}} is isomorphic to&nbsp;{{tmath|\Z_p.}}

Similarly, the formal power series ring {{math|''R''[{[''t'']}]}} is the completion of {{math|''R''[''t'']}} at {{math|(''t'')}} (see also ''[[Hensel's lemma]]'')<!-- need to be discussed with concrete examples. Start of a para: The notion of the completion formalizes the construction of a solution by successive approximation. -->

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 {{mvar|F}} be a [[free ring]] (that is, free algebra over the integers) with the set {{mvar|X}} of symbols, that is, {{mvar|F}} consists of polynomials with integral coefficients in noncommuting variables that are elements of {{mvar|X}}. A free ring satisfies the universal property: any function from the set {{mvar|X}} to a ring {{mvar|R}} factors through {{mvar|F}} so that {{math|''F'' → ''R''}} is the unique ring homomorphism. Just as in the group case, every ring can be represented as a quotient of a free ring.{{sfnp|Cohn|1995|loc=[https://books.google.com/books?id=u-4ADgUgpSMC&pg=PA242 pg. 242]|ps=}}

Now, we can impose relations among symbols in {{mvar|X}} by taking a quotient. Explicitly, if {{mvar|E}} is a subset of {{mvar|F}}, then the quotient ring of {{mvar|F}} by the ideal generated by {{mvar|E}} is called the ring with generators {{mvar|X}} and relations {{mvar|E}}. If we used a ring, say, {{mvar|A}} as a base ring instead of {{tmath|\Z,}} then the resulting ring will be over {{mvar|A}}. For example, if <math>E = \{ xy - yx \mid x, y \in X \},</math> then the resulting ring will be the usual polynomial ring with coefficients in {{mvar|A}} in variables that are elements of {{mvar|X}} (It is also the same thing as the [[symmetric algebra]] over {{mvar|A}} with symbols {{mvar|X}}.)

In the category-theoretic terms, the formation <math>S \mapsto \text{the free ring generated by the set } S</math> 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.)<!-- Note: we need to discuss a ring as a solution to a universal problem with some specific application. -->

Let {{math|''A''}}, {{math|''B''}} be algebras over a commutative ring {{mvar|R}}. Then the tensor product of {{mvar|R}}-modules <math>A \otimes_R B</math> is an {{mvar|R}}-algebra with multiplication characterized by <math>(x \otimes u) (y \otimes v) = xy \otimes uv.</math> {{See also|Tensor product of algebras|Change of rings}}