polynomial ring

《代数学引论》 · 第 118 页

我们归纳地定义 $R$ 上 $n$ 个未定元 $x_1,\dots,x_n$ 的多项式环（记作 $R[x_1,\dots,x_n]$）如下：当 $n=1$ 时 $R[x_1]$ 已在上面定义（定义 7）。

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《Introduction to Geometry (Second Edition)》 · 第 19 页 · Chapter 1，Section 1.3，Definition 1.48

The polynomials $f(x)=\sum_{i=0}^n a_ix^i$ and $g(x)=\sum_{i=0}^n b_ix^i$ over $R$ are considered equal if and only if $a_i=b_i$ for $0\leq i\leq n$. We define the sum of $f(x)$ and $g(x)$ by $f(x)+g(x)=\sum_{i=0}^n(a_i+b_i)x^i$. To define the product of two polynomials over $R$, let $f(x)=\sum_{i=0}^n a_ix^i$ and $g(x)=\sum_{j=0}^m b_jx^j$ and set $f(x)g(x)=\sum_{k=0}^{n+m}c_kx^k$, where $c_k=\sum_{\substack{i+j=k\\0\leq i\leq n,\ 0\leq j\leq m}}a_ib_j$. It is easily seen that with these operations the set of polynomials over $R$ forms a ring. The ring formed by the polynomials over $R$ with the above operations is called the polynomial ring over $R$ and denoted by $R[x]$.