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# 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 $(a_{0},a_{1},a_{2},a_{3},\dots)$ is an element in $R[X]$, with $a_{n}=0$ for all $n>N$, then we usually write this element as

$\sum_{{n=0}}^{N}a_{n}X^{n}=a_{0}+a_{1}X+a_{2}X^{2}+a_{3}X^{3}+\cdots+a_{N}X^{N}.$ |

Elements of $R[X]$ are called *polynomials* in the indeterminate $X$ with coefficients in $R$. The ring elements $a_{0},\ldots,a_{N}$ are called *coefficients* of the polynomial, and the *degree* of a polynomial is the largest natural number $N$ for which $a_{N}\neq 0$, 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 $-\infty$.

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

$\displaystyle\sum_{{n=0}}^{N}a_{n}X^{n}+\sum_{{n=0}}^{N}b_{n}X^{n}$ | $\displaystyle=$ | $\displaystyle\sum_{{n=0}}^{N}(a_{n}+b_{n})X^{n}$ | (1) | ||

$\displaystyle\sum_{{n=0}}^{N}a_{n}X^{n}\cdot\sum_{{n=0}}^{N}b_{n}X^{n}$ | $\displaystyle=$ | $\displaystyle\sum_{{n=0}}^{{2N}}\left(\sum_{{k=0}}^{n}a_{k}b_{{n-k}}\right)X^{n}$ | (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 $R[X,Y,Z]:=R[X,Y][Z]=R[X][Y][Z]\cong R[X][Z][Y]\cong\cdots$, and in any finite number of variables, we have inductively $R[X_{1},X_{2},\dots,X_{n}]:=R[X_{1},\dots,X_{{n-1}}][X_{n}]=R[X_{1}][X_{2}]% \cdots[X_{n}]$, 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 $A=\{a_{1},\ldots,a_{n}\}$ of $E(M)$, set $R[A]:=R[a_{1},\ldots,a_{n}]$. 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 $R[M]$.

## Mathematics Subject Classification

11C08*no label found*12E05

*no label found*13P05

*no label found*17B66

*no label found*16W10

*no label found*70G65

*no label found*17B45

*no label found*

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