# polynomial ring

## 1 Polynomial rings in one variable

Let $R$ be a ring. The *polynomial ring ^{}* over $R$ in one variable $X$ is the set $R[X]$ of all sequences

^{}in $R$ with only finitely many nonzero terms. If $({a}_{0},{a}_{1},{a}_{2},{a}_{3},\mathrm{\dots})$ is an element in $R[X]$, with ${a}_{n}=0$ for all $n>N$, then we usually write this element as

$$\sum _{n=0}^{N}{a}_{n}{X}^{n}={a}_{0}+{a}_{1}X+{a}_{2}{X}^{2}+{a}_{3}{X}^{3}+\mathrm{\cdots}+{a}_{N}{X}^{N}.$$ |

Elements of $R[X]$ are called *polynomials ^{}* in the indeterminate $X$ with coefficients in $R$. The ring elements ${a}_{0},\mathrm{\dots},{a}_{N}$ are called

*coefficients*of the polynomial, and the

*degree*of a polynomial is the largest natural number

^{}$N$ for which ${a}_{N}\ne 0$, if such an $N$ exists. When a polynomial has all of its coefficients equal to $0$, its degree is usually considered to be undefined, although some people adopt the convention that its degree is $-\mathrm{\infty}$.

A *monomial* is a polynomial with exactly one nonzero coefficient. Similarly, a *binomial* is a polynomial with exactly two nonzero coefficients, and a *trinomial* is a polynomial with exactly three nonzero coefficients.

Addition and multiplication of polynomials is defined by

$\sum _{n=0}^{N}}{a}_{n}{X}^{n}+{\displaystyle \sum _{n=0}^{N}}{b}_{n}{X}^{n$ | $=$ | $\sum _{n=0}^{N}}({a}_{n}+{b}_{n}){X}^{n$ | (1) | ||

$\sum _{n=0}^{N}}{a}_{n}{X}^{n}\cdot {\displaystyle \sum _{n=0}^{N}}{b}_{n}{X}^{n$ | $=$ | $\sum _{n=0}^{2N}}\left({\displaystyle \sum _{k=0}^{n}}{a}_{k}{b}_{n-k}\right){X}^{n$ | (2) |

$R[X]$ is a $\mathbb{Z}$–graded ring^{} under these operations^{}, with the monomials of degree exactly $n$ comprising the ${n}^{\mathrm{th}}$ graded component^{} of $R[X]$. The zero element^{} of $R[X]$ is the polynomial whose coefficients are all $0$, and when $R$ has a multiplicative identity^{} $1$, the polynomial whose coefficients are all $0$ except for ${a}_{0}=1$ is a multiplicative identity for the polynomial ring $R[X]$.

## 2 Polynomial rings in finitely many variables

The *polynomial ring* over $R$ in two variables $X,Y$ is defined to be $R[X,Y]:=R[X][Y]\cong R[Y][X]$. Elements of $R[X,Y]$ are called *polynomials* in the indeterminates $X$ and $Y$ with coefficients in $R$. A *monomial* in $R[X,Y]$ is a polynomial which is simultaneously a monomial in both $X$ and $Y$, when considered as a polynomial in $X$ with coefficients in $R[Y]$ (or as a polynomial in $Y$ with coefficients in $R[X]$). The *degree* of a monomial in $R[X,Y]$ is the sum of its individual degrees in the respective indeterminates $X$ and $Y$ (in $R[Y][X]$ and $R[X][Y]$), and the degree of a polynomial in $R[X,Y]$ is the supremum of the degrees of its monomial summands, if it has any.

In three variables, we have $R[X,Y,Z]:=R[X,Y][Z]=R[X][Y][Z]\cong R[X][Z][Y]\cong \mathrm{\cdots}$, and in any finite number of variables, we have inductively $R[{X}_{1},{X}_{2},\mathrm{\dots},{X}_{n}]:=R[{X}_{1},\mathrm{\dots},{X}_{n-1}][{X}_{n}]=R[{X}_{1}][{X}_{2}]\mathrm{\cdots}[{X}_{n}]$, with monomials and degrees defined in analogy^{} to the two variable case. In any number of variables, a polynomial ring is a graded ring with ${n}^{\mathrm{th}}$ graded component equal to the $R$-module generated by the monomials of degree $n$.

## 3 Polynomial rings in arbitrarily many variables

For any nonempty set $M$, let $E(M)$ denote the set of all finite subsets of $M$. For each element $A=\{{a}_{1},\mathrm{\dots},{a}_{n}\}$ of $E(M)$, set $R[A]:=R[{a}_{1},\mathrm{\dots},{a}_{n}]$. Any two elements $A,B\in E(M)$ satisfying $A\subset B$ give rise to the relationship $R[A]\subset R[B]$ if we consider $R[A]$ to be embedded in $R[B]$ in the obvious way. The union of the rings $\{R[A]:A\in E(M)\}$ (or, more formally, the categorical direct limit^{} of the direct system^{} of rings $\{R[A]:A\in E(M)\}$) is defined to be the ring $R[M]$.

Title | polynomial ring |

Canonical name | PolynomialRing |

Date of creation | 2013-03-22 11:52:27 |

Last modified on | 2013-03-22 11:52:27 |

Owner | djao (24) |

Last modified by | djao (24) |

Numerical id | 10 |

Author | djao (24) |

Entry type | Definition |

Classification | msc 11C08 |

Classification | msc 12E05 |

Classification | msc 13P05 |

Classification | msc 17B66 |

Classification | msc 16W10 |

Classification | msc 70G65 |

Classification | msc 17B45 |

Related topic | AlgebraicGeometry |

Related topic | RationalFunction |

Defines | polynomial |

Defines | monomial |

Defines | binomial |

Defines | trinomial |

Defines | degree |