Editing Ring (mathematics) (section)

=== 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]]).