Ring of polynomials

polynomial ring

A ring whose elements are polynomials (cf. Polynomial) with coefficients in some fixed field . Rings of polynomials over an arbitrary commutative associative ring , for example, over the ring of integers, are also discussed. The accepted notation for the ring of polynomials in a finite set of variables over is . 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 is a (commutative) free algebra with an identity over ; 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 there is Hilbert's basis theorem: Every ideal in is finitely generated (as an ideal) (cf. Hilbert theorem). A ring of polynomials in one variable over a field, is a principal ideal ring, that is, each ideal of it is generated by one element. Moreover, is a Euclidean ring. This property of 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 the ring is not a principal ideal ring.

Let be a commutative associative -algebra with an identity, and let be an element of the Cartesian power . Then there is a unique -algebra homomorphism of the ring of polynomials in variables into ,

for which , for all , and is the identity of . The image of a polynomial under this homomorphism is called its value at the point . A point is called a zero of a system of polynomials if the value of each polynomial from at this point is . For a ring of polynomials there is Hilbert's Nullstellen Satz: Let be an ideal in the ring , let be the set of zeros of in , where is the algebraic closure of , and let be a polynomial in vanishing at all points of . Then there is a natural number such that (cf. Hilbert theorem).

Let be an arbitrary module over the ring . Then there are free -modules and homomorphisms such that the sequence of homomorphisms

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 generated by some set for which is a non-zero constant? 3) If is isomorphic to , must be isomorphic to ?

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