域，环

域

definition of fields

域（field）是一种代数结构（algebraic structure）。

... This may be summarized by saying: a field has two operations, the addition and the multiplication; it is an abelian group under addition, with $0$ as additive identity; the nonzero elements form an abelian group under multiplication (with $1$ as multiplicative identity), and the multiplication is distributive over addition.
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数域

复数域的子域是为数域。

例子

模2域 $\{0,1\}$，满足 $0+0=0, 0 + 1 = 1, 1+1 = 0, 0\times 0 = 0, 0 \times 1 = 0, 1 \times 1 = 1$ 。

环

A ring is a set $R$ equipped with two binary operations addition and multiplication satisfying the following three sets of axioms, called the ring axioms

1. $R$ is an abelian group under addition,
2. $R$ is a monoid（幺半群）under multiplication,
3. Multiplication is distributive with respect to addition.

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疑问：「整环」的英文是「integral domain」但是「环」字按理说应该跟「ring」对应，那么「domain」究竟是什么意思呢？

TO-DO:

• Euclidean Ring

转载于:https://www.cnblogs.com/Patt/p/9141570.html