域
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.
SOURCE
数域
复数域的子域是为数域。
例子
模2域 ${0,1}$，满足 $0+0=0, 0 + 1 = 1, 1+1 = 0, 0 imes 0 = 0, 0 imes 1 = 0, 1 imes 1 = 1$ 。
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
- $R$ is an abelian group under addition,
- $R$ is a monoid（幺半群）under multiplication,
- Multiplication is distributive with respect to addition.
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