Ring of polynomials

polynomial ring

A ring whose elements are polynomials 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 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 [5], [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

[1]	S. Lang, "Algebra" , Addison-Wesley (1974)
[2]	N. Bourbaki, "Algèbre" , Eléments de mathématiques , 2 , Masson (1981) pp. Chapts. 4; 5; 6
[3]	D. Hilbert, "Ueber die vollen Invariantensysteme" Math. Ann. , 42 (1893) pp. 313–373
[4]	D. Hilbert, "Ueber die Theorie der algebraischen Formen" Math. Ann. , 36 (1890) pp. 473–534
[5]	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
[6]	D. Quillen, "Projective modules over polynomial rings" Invent. Math. , 36 (1976) pp. 167–171

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

[a1]	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

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 $M_X$.