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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 \begin{eqnarray} \sum_{n=0}^N a_n X^n + \sum_{n=0}^N b_n X^n & = & \sum_{n=0}^N (a_n+b_n) X^n \\ \sum_{n=0}^N a_n X^n \cdot \sum_{n=0}^N b_n X^n & = & \sum_{n=0}^{2N} \left(\sum_{k=0}^n a_k b_{n-k}\right) X^n \end{eqnarray}$R[X]$ 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 $R[X]$ .
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 $n$ .
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 $R[M]$ .
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