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''polynomial ring''

A [[ring]] whose elements are [[polynomial]]s with coefficients in some fixed [[field]] $k$. Rings of polynomials over an arbitrary commutative associative ring $R$, for example, over the ring of integers, are also discussed. The accepted notation for the ring of polynomials in a finite set of variables $x_1,\ldots,x_n$ over $R$ is $R[x_1,\ldots,x_n]$. It is possible to speak of a ring of polynomials in an infinite set of variables if it is assumed that each individual polynomial depends only on a finite number of variables. A ring of polynomials over a ring $R$ is a (commutative) [[free algebra]] with an identity over $R$; the set of variables serves as a system of free generators of this algebra.

A ring of polynomials over an arbitrary [[integral domain]] is itself an integral domain. A ring of polynomials over a [[factorial ring]] is itself factorial.

For a ring of polynomials in a finite number of variables over a field $k$ there is Hilbert's basis theorem: Every ideal in $k[x_1,\ldots,x_n]$ is finitely generated (as an ideal) (cf. [[Hilbert theorem]]). A ring of polynomials in one variable over a field, $k[x]$ is a [[principal ideal ring]], that is, each ideal of it is generated by one element. Moreover, $k[x]$ is a [[Euclidean ring]]. This property of $k[x]$ gives one the possibility of comprehensively describing the finitely-generated modules over it and, in particular, of reducing linear operators in a finite-dimensional vector space to canonical form (see [[Jordan matrix]]). For $n>1$ the ring $k[x_1,\ldots,x_n]$ is not a principal ideal ring.

Let $S$ be a commutative associative $k$-algebra with an identity, and let $a = (a_1,\ldots,a_n)$ be an element of the Cartesian power $S^n$. Then there is a unique $k$-algebra homomorphism of the ring of polynomials in $n$ variables into $S$,
$$
\phi_a : k[x_1,\ldots,x_n] \rightarrow S
$$
for which $\phi(x_i) = a_i$, for all $i = 1,\ldots,n$, and $\phi_a(1)$ is the identity of $S$. The image of a polynomial $f \in k[x_1,\ldots,x_n]$ under this homomorphism is called its value at the point $a$. A point $a \in S^n$ is called a zero of a system of polynomials $F \subset k[x_1,\ldots,x_n]$ if the value of each polynomial from $F$ at this point is $0$. For a ring of polynomials there is Hilbert's [[Nullstellen Satz]]: Let $\mathfrak{A}$ be an ideal in the ring $R = k[x_1,\ldots,x_n]$, let $M$ be the set of zeros of $\mathfrak{A}$ in $\bar k^n$, where $\bar k$ is the algebraic closure of $k$, and let $g$ be a polynomial in $R$ vanishing at all points of $M$. Then there is a natural number $m$ such that $g^m \in \mathfrak{A}$ (cf. [[Hilbert theorem]]).

Let $A$ be an arbitrary module over the ring $R = k[x_1,\ldots,x_n]$. Then there are free $R$-modules $X_0,\ldots,X_n$ and homomorphisms $X_i \rightarrow x_{i-1}$ such that the sequence of homomorphisms
$$
0 \leftarrow A \leftarrow X_0 \leftarrow \cdots \leftarrow X_n \leftarrow 0
$$
is exact, that is, the kernel of one homomorphism is the image of the next. This result is one possible formulation of the [[Hilbert theorem]] on [[Syzygy|syzygies]] for a ring of polynomials.

A finitely-generated projective module over a ring of polynomials in a finite number of variables with coefficients from a principal ideal ring is free (see [[#References|[5]]], [[#References|[6]]]); this is the solution of Serre's problem.

Only in certain particular cases are there answers to the following questions: 1) Is the group of automorphisms of a ring of polynomials generated by elementary automorphisms? 2) Is $k[x_1,\ldots,x_n]$ generated by some set $f_1,\ldots,f_n$ for which $\det|\partial f_i / \partial x_j|$ is a non-zero constant? 3) If $S \otimes k[y]$ is isomorphic to $k[x_1,\ldots,x_n]$, must $S$ be isomorphic to $k[x_1,\ldots,x_{n-1}]$?

====References====
<table>
<TR><TD valign="top">[1]</TD> <TD valign="top">  S. Lang,   "Algebra" , Addison-Wesley  (1974)</TD></TR>
<TR><TD valign="top">[2]</TD> <TD valign="top">  N. Bourbaki,   "Algèbre" , ''Eléments de mathématiques'' , '''2''' , Masson  (1981)  pp. Chapts. 4; 5; 6</TD></TR>
<TR><TD valign="top">[3]</TD> <TD valign="top">  D. Hilbert,   "Ueber die vollen Invariantensysteme"  ''Math. Ann.'' , '''42'''  (1893)  pp. 313–373</TD></TR>
<TR><TD valign="top">[4]</TD> <TD valign="top">  D. Hilbert,   "Ueber die Theorie der algebraischen Formen"  ''Math. Ann.'' , '''36'''  (1890)  pp. 473–534</TD></TR>
<TR><TD valign="top">[5]</TD> <TD valign="top">  A.A. Suslin,   "Projective modules over a polynomial ring are free"  ''Soviet Math. Dokl.'' , '''17''' :  4  (1976)  pp. 1160–1164  ''Dokl. Akad. Nauk SSSR'' , '''229'''  (1976)  pp. 1063–1066</TD></TR>
<TR><TD valign="top">[6]</TD> <TD valign="top">  D. Quillen,   "Projective modules over polynomial rings"  ''Invent. Math.'' , '''36'''  (1976)  pp. 167–171</TD></TR>
</table>

====Comments====
The following properties of a ring $R$ lift to the polynomial ring $R[x]$, and hence to the ring $R[x_1,\ldots,x_n]$: $R$ is an [[integral domain]]; $R$ is a [[unique factorisation domain]]; $R$ is a [[Noetherian ring]]; $R$ is a [[normal ring]].

====References====
<table>
<TR><TD valign="top">[a1]</TD> <TD valign="top">  Nagata, Masayoshi ''Polynomial rings and affine spaces'' CBMS Regional Conference Series in Mathematics '''37''' American Mathematical Society (1978) {{ISBN|0-8218-1687-X}} {{ZBL|0391.14001}}</TD></TR>
</table>

====Comment====
The "variables" $x_i$ may also be referred to as ''indeterminates'' or ''transcendentals'' over the base ring $R$.  The term should not be taken to indicate that a polynomial is to be identified with the function that it defines on $R^n$: over the [[finite field]] $\mathbb{F}_p$, for example, the polynomials $X$ and $X^p$ are distinct but define the same function.

For a given set of indeterminates $X = \{x_1,\ldots,x_n\}$, the monomials may be identified with the elements of the free commutative monoid $M_X$ on the alphabet $X$, and the polynomial ring is then the [[monoid ring]] of $M_X$.

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