封闭性：a + b is another element in the set
结合律：(a + b) + c = a + (b + c)
单位元：加法的单位元为0，a + 0 = a and 0 + a = a
逆 元：加法的逆元为-a，a + (−a) = (−a) + a = 0 (对于所有元素)
交换律：a + b = b + a
(2)(R, ·)是幺半群
结合律：(a ⋅ b) ⋅ c = a ⋅ (b ⋅ c)
单位元：乘法的单位元为1，a ⋅ 1 = a and 1 ⋅ a = a
(3)乘法对加法满足分配律Multiplication distributes over addition
a ⋅ (b + c) = (a ⋅ b) + (a ⋅ c) for all a, b, c in R (left distributivity)
(b + c) ⋅ a = (b ⋅ a) + (c ⋅ a) for all a, b, c in R (right distributivity)
3.1 环 环(ring)在阿贝尔群(也叫交换群)的基础上，添加一种二元运算·(虽叫乘法，但不同于初等代数的乘法)。一个代数结构是环(R, +, ·)，需要满足环公理(ring axioms)，如(Z,+, ⋅)。环公理如下：
(1)(R, +)是交换群
封闭性：a + b is another element in the set 结合律：(a + b) + c = a + (b + c) 单位元：加法的单位元为0，a + 0 = a and 0 + a = a 逆 元：加法的逆元为-a，a + (−a) = (−a) + a = 0 (对于所有元素) 交换律：a + b = b + a (2)(R, ·)是幺半群
结合律：(a ⋅ b) ⋅ c = a ⋅ (b ⋅ c) 单位元：乘法的单位元为1，a ⋅ 1 = a and 1 ⋅ a = a (3)乘法对加法满足分配律Multiplication distributes over addition
a ⋅ (b + c) = (a ⋅ b) + (a ⋅ c) for all a, b, c in R (left distributivity) (b + c) ⋅ a = (b ⋅ a) + (c ⋅ a) for all a, b, c in R (right distributivity)