# Polynomial ring

A polynomial ring is a ring whose elements are polynomials, that is, finite formal sums of coefficients from a base ring multiplied by powers of one or more indeterminates such as \( x \). For a commutative ring \( R \) with identity, the polynomial ring in one indeterminate is written \( R[x] \), and in \( n \) indeterminates \( R[x_1, \ldots, x_n] \). Encyclopedia of Mathematics defines it as a free commutative algebra with identity over \( R \), with the set of variables serving as a system of free generators.<sup>[1](https://encyclopediaofmath.org/wiki/Polynomial_ring)</sup>

| Key fact | Detail |
|---|---|
| Elements | Finite sums \( f(x) = a_0 + a_1 x + \cdots + a_n x^n \) with coefficients \( a_i \in R \); the coefficient \( a_n \neq 0 \) of the highest power is the leading coefficient, and a polynomial with leading coefficient 1 is monic.<sup>[2](https://math.libretexts.org/Bookshelves/Abstract_and_Geometric_Algebra/Abstract_Algebra%3A_Theory_and_Applications_%28Judson%29/17%3A_Polynomials/17.01%3A_Polynomial_Rings)</sup> |
| Structural description | \( R[x_1, \ldots, x_n] \) is the monoid algebra of the free commutative monoid \( \mathbb{N}^n \); monomials \( x_1^{a_1} \cdots x_n^{a_n} \) form a basis as an \( R \)-module.<sup>[3](https://www.cip.ifi.lmu.de/~grinberg/t/21w/lecb.pdf)</sup><sup> • </sup><sup>[1](https://encyclopediaofmath.org/wiki/Polynomial_ring)</sup> |
| Multivariate construction | Built by iterating the one-variable construction: \( R[x_1, \ldots, x_m] = R[x_1, \ldots, x_{m-1}][x_m] \).<sup>[4](https://tim4datfau.github.io/Timothy-Ford-at-FAU/preprints/Algebra_Book_1.pdf)</sup> |
| Hilbert's basis theorem (1890) | Every ideal in \( k[x_1, \ldots, x_n] \) over a field \( k \) is finitely generated; the Noetherian condition passes from \( R \) to \( R[x] \).<sup>[1](https://encyclopediaofmath.org/wiki/Polynomial_ring)</sup><sup> • </sup><sup>[5](https://syntagmatic.github.io/moonshine/noether/12-hilberts-basis-theorem.html)</sup> |
| Properties that lift | If \( R \) is an integral domain, a unique factorization domain, a Noetherian ring, or a normal ring, then so are \( R[x] \) and \( R[x_1, \ldots, x_n] \).<sup>[1](https://encyclopediaofmath.org/wiki/Polynomial_ring)</sup> |
| Polynomials vs. functions | A polynomial is not the function it defines: over the finite field \( \mathbb{F}_p \), the polynomials \( X \) and \( X^p \) are distinct but define the same function.<sup>[1](https://encyclopediaofmath.org/wiki/Polynomial_ring)</sup> |
| Landmark result | Quillen and Suslin independently proved in 1976 that every finitely generated projective module over a polynomial ring in finitely many variables over a field is free.<sup>[1](https://encyclopediaofmath.org/wiki/Polynomial_ring)</sup><sup> • </sup><sup>[11](https://link.springer.com/chapter/10.1007/978-3-0348-9996-3_3)</sup> |

## Definition and basic structure

Throughout the standard treatment, \( R \) is a commutative ring with identity. A polynomial over \( R \) in the indeterminate \( x \) is an expression \( f(x) = \sum_{i=0}^{n} a_i x^i \) with \( a_i \in R \) and \( a_n \neq 0 \). The elements \( a_0, a_1, \ldots, a_n \) are the coefficients; \( a_n \) is the leading coefficient, a polynomial with leading coefficient 1 is monic, and the degree of \( f \) is the largest \( n \) with \( a_n \neq 0 \).<sup>[2](https://math.libretexts.org/Bookshelves/Abstract_and_Geometric_Algebra/Abstract_Algebra%3A_Theory_and_Applications_%28Judson%29/17%3A_Polynomials/17.01%3A_Polynomial_Rings)</sup>

**Formal sums, not functions.** The indeterminates \( x_i \) are also called transcendentals over the base ring, and the term should not be taken to mean that a polynomial is the function it defines on \( R^n \). Over \( \mathbb{F}_p \), \( X \) and \( X^p \) are distinct polynomials that agree at every point of the field.<sup>[1](https://encyclopediaofmath.org/wiki/Polynomial_ring)</sup>

A precise modern definition identifies monomials with the elements of the free commutative monoid on the alphabet \( X = \{x_1, \ldots, x_n\} \), so that the polynomial ring is the monoid ring of that monoid.<sup>[1](https://encyclopediaofmath.org/wiki/Polynomial_ring)</sup> Concretely, \( R[x_1, \ldots, x_n] \) is a free \( R \)-module with basis the monomials \( x_1^{a_1} \cdots x_n^{a_n} \) for exponent tuples \( (a_1, \ldots, a_n) \in \mathbb{N}^n \), and every polynomial is uniquely an \( R \)-linear combination of finitely many of them.<sup>[3](https://www.cip.ifi.lmu.de/~grinberg/t/21w/lecb.pdf)</sup> The ring \( R \) itself sits inside as the constant polynomials, identified with elements \( r \cdot 1 \).<sup>[3](https://www.cip.ifi.lmu.de/~grinberg/t/21w/lecb.pdf)</sup>

**Several variables.** The multivariate ring is defined by iterating the one-variable construction, \( R[x_1, \ldots, x_m] = R[x_1, \ldots, x_{m-1}][x_m] \), and the order of adjoining variables does not matter: \( (R[x])[y] \cong (R[y])[x] \), so the notation \( R[x, y] \) is unambiguous.<sup>[4](https://tim4datfau.github.io/Timothy-Ford-at-FAU/preprints/Algebra_Book_1.pdf)</sup><sup> • </sup><sup>[2](https://math.libretexts.org/Bookshelves/Abstract_and_Geometric_Algebra/Abstract_Algebra%3A_Theory_and_Applications_%28Judson%29/17%3A_Polynomials/17.01%3A_Polynomial_Rings)</sup>

## Evaluation

Substituting an element for the indeterminate gives a ring homomorphism. For \( \alpha \in R \), the map \( \phi_{\alpha} : R[x] \to R \) defined by \( \phi_{\alpha}(p(x)) = p(\alpha) = a_n \alpha^n + \cdots + a_1 \alpha + a_0 \) is a ring homomorphism.<sup>[2](https://math.libretexts.org/Bookshelves/Abstract_and_Geometric_Algebra/Abstract_Algebra%3A_Theory_and_Applications_%28Judson%29/17%3A_Polynomials/17.01%3A_Polynomial_Rings)</sup> [Evaluation](https://www.edgepedia.org/evaluation) is more general than substitution into \( R \) itself: a univariate polynomial over \( R \) can be evaluated at any element of any \( R \)-algebra, including a square matrix ring over \( R \) or the polynomial ring \( R[x] \) itself.<sup>[3](https://www.cip.ifi.lmu.de/~grinberg/t/21w/lecb.pdf)</sup>

## Ideals and the Hilbert basis theorem

The ring's most consequential property is that its ideals are finitely generated. [Hilbert's basis theorem](https://www.edgepedia.org/hilberts-basis-theorem) states that for a field \( k \), every ideal in \( k[x_1, \ldots, x_n] \) is finitely generated as an ideal.<sup>[1](https://encyclopediaofmath.org/wiki/Polynomial_ring)</sup> The mechanism is the Noetherian chain condition passing from a ring to the polynomials over it: applied one variable at a time, it shows that any polynomial ring in finitely many variables over a field or over \( \mathbb{Z} \) is Noetherian.<sup>[5](https://syntagmatic.github.io/moonshine/noether/12-hilberts-basis-theorem.html)</sup> In an 1890 paper, Hilbert applied the theorem to show that the invariants of forms in arbitrarily many variables form a finitely generated set, and because his proof established this without giving explicit constructions, [Paul Gordan](https://www.edgepedia.org/paul-gordan) is said to have objected, "Das ist nicht Mathematik, das ist Theologie"; within a few years Hilbert supplied a more constructive approach.<sup>[5](https://syntagmatic.github.io/moonshine/noether/12-hilberts-basis-theorem.html)</sup>

The ideal theory of polynomial rings developed through the work of Hilbert, Lasker, and Macauley in 1890, 1905, and 1913. Lasker's major result was primary decomposition: every ideal in a polynomial ring \( F[x_1, \ldots, x_n] \) is a finite intersection of primary ideals, and Macauley proved that the associated primes of a minimal such decomposition are uniquely determined, although the embedded primary components need not be unique, while the components corresponding to isolated primes are unique.<sup>[6](https://www.researchgate.net/publication/225767243_From_Numbers_to_Rings_The_Early_History_of_Ring_Theory)</sup><sup> • </sup><sup>[12](https://www.math.ucla.edu/~sharifi/notes/algebra-ch11.html)</sup> [Emmy Noether](https://www.edgepedia.org/emmy-noether)'s 1921 *Idealtheorie in Ringbereichen* recast the basis theorem as the chain-condition argument used in modern proofs.<sup>[5](https://syntagmatic.github.io/moonshine/noether/12-hilberts-basis-theorem.html)</sup>

## Factorization properties

Several ring-theoretic properties lift from the coefficient ring to the polynomial ring: if \( R \) is an integral domain, a unique factorization domain, a [Noetherian ring](https://www.edgepedia.org/noetherian-ring), or a normal ring, then so are \( R[x] \) and \( R[x_1, \ldots, x_n] \).<sup>[1](https://encyclopediaofmath.org/wiki/Polynomial_ring)</sup> When the coefficient ring has zero divisors the situation changes, and the literature characterizes exactly when a polynomial ring over an arbitrary commutative ring has unique factorization.<sup>[7](https://arxiv.org/abs/1906.00522)</sup>

A famous problem in this area was settled in 1976, when [Daniel Quillen](https://www.edgepedia.org/daniel-quillen) and [Andrei Suslin](https://www.edgepedia.org/andrei-suslin) independently proved that every finitely generated projective module over a polynomial ring in finitely many variables over a field is free, confirming a question related to Serre's problem.<sup>[1](https://encyclopediaofmath.org/wiki/Polynomial_ring)</sup>

## Connection to algebraic geometry

Polynomial rings carry the algebra side of algebraic geometry. If \( V \) is a variety, the set of polynomials vanishing on it, \( I(V) = \{ f \in R[x_1, \ldots, x_n] : f(a_1, \ldots, a_n) = 0 \text{ for all } (a_1, \ldots, a_n) \in V \} \), is an ideal; conversely, an ideal \( J \) defines a variety \( V(J) \) of common zeros. Over an algebraically closed field this correspondence becomes a one-to-one correspondence between varieties and their largest defining ideals, the radical ideals, with prime ideals corresponding to irreducible varieties.<sup>[6](https://www.researchgate.net/publication/225767243_From_Numbers_to_Rings_The_Early_History_of_Ring_Theory)</sup>

**Computation.** The ring \( k[x_1, \ldots, x_n] \) also supports algorithmic methods. A division algorithm in \( k[x_1, \ldots, x_n] \), monomial ideals, Dickson's lemma, the Hilbert basis theorem, and Gröbner bases form the computational core, followed by [Buchberger's algorithm](https://www.edgepedia.org/buchbergers-algorithm), elimination and extension theorems, and the Nullstellensatz.<sup>[8](https://ocw.mit.edu/courses/18-704-seminar-in-algebra-and-number-theory-computational-commutative-algebra-and-algebraic-geometry-fall-2008/pages/readings/)</sup> Monomial orders are central here: as with [Gaussian elimination](https://www.edgepedia.org/gaussian-elimination) for linear systems, an ordering of terms is a key ingredient of the division algorithm, and Gröbner bases allow problems about polynomial ideals to be solved algorithmically.<sup>[9](https://www.ms.uky.edu/~corso/teaching/math764/syllabus.html)</sup> [Computer algebra](https://www.edgepedia.org/computer-algebra) systems reflect this design: Magma can create a multivariate polynomial ring \( P = R[x_1, \ldots, x_n] \) over an arbitrary coefficient ring, stores polynomials in distributive form as arrays of coefficient-monomial pairs, and separates monomial orders from basic arithmetic because orders affect Gröbner basis computations but not the ring operations themselves.<sup>[10](https://docs.magma-maths.org/BasicRings/MultivariatePolynomialRings/introduction.html)</sup>

## References

1. [Polynomial ring, Encyclopedia of Mathematics](https://encyclopediaofmath.org/wiki/Polynomial_ring)
2. [17.1: Polynomial Rings, Thomas W. Judson, Abstract Algebra: Theory and Applications, Mathematics LibreTexts](https://math.libretexts.org/Bookshelves/Abstract_and_Geometric_Algebra/Abstract_Algebra%3A_Theory_and_Applications_%28Judson%29/17%3A_Polynomials/17.01%3A_Polynomial_Rings)
3. [Deyuan Guo / Darij Grinberg, Lecture 11: Polynomial rings, LMU Munich (February 2023)](https://www.cip.ifi.lmu.de/~grinberg/t/21w/lecb.pdf)
4. [Timothy Ford, Rings (algebra text chapter, Florida Atlantic University)](https://tim4datfau.github.io/Timothy-Ford-at-FAU/preprints/Algebra_Book_1.pdf)
5. [Hilbert's Basis Theorem, Invariance: The Mathematics of Emmy Noether](https://syntagmatic.github.io/moonshine/noether/12-hilberts-basis-theorem.html)
6. [Israel Kleiner, From Numbers to Rings: The Early History of Ring Theory, Elemente der Mathematik 53 (1998)](https://www.researchgate.net/publication/225767243_From_Numbers_to_Rings_The_Early_History_of_Ring_Theory)
7. [Unique Factorization in Polynomial Rings with Zero Divisors, arXiv:1906.00522](https://arxiv.org/abs/1906.00522)
8. [Readings, 18.704 Seminar in Computational Commutative Algebra and Algebraic Geometry, MIT OpenCourseWare (Fall 2008)](https://ocw.mit.edu/courses/18-704-seminar-in-algebra-and-number-theory-computational-commutative-algebra-and-algebraic-geometry-fall-2008/pages/readings/)
9. [Course syllabus, Math 764 Geometry, Algebra & Algorithms, University of Kentucky](https://www.ms.uky.edu/~corso/teaching/math764/syllabus.html)
10. [Introduction, Multivariate Polynomial Rings, Magma Handbook](https://docs.magma-maths.org/BasicRings/MultivariatePolynomialRings/introduction.html)
11. [Client Challenge](https://link.springer.com/chapter/10.1007/978-3-0348-9996-3_3)
12. [Abstract Algebra, Chapter 11 Commutative algebra · Romyar Sharifi](https://www.math.ucla.edu/~sharifi/notes/algebra-ch11.html)

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*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Numbers and algebra › Algebraic structures › Ring theory › Commutative algebra › Polynomial and power-series rings over commutative rings*

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