Totui

定義:

Let R be a non-empty set on which we have two closed binary operations, denoted by + and 

(which may be quite different from the ordinary addition and multiplication to which we are accustomed).

Then (R, +,   ) is a ring if for all a, b, c  R , the following conditions are satisifed:

斜體R代表實數(Real Number)

Commutative Law of + (+交換律) : a + b = b + a

Associative Law of + (+結合律) :  a + (b + c) = (a + b) + c

Existence of an identity for + (+, 存在原點) : There exists z  R  such that, a + z = z + a = a for every a  R 

例: a + 0 = 0 + a = a,  0就是一般+號的原點

Existence of inverses under +(+, 存在反元素): For each a  R there is an element b  R with a + b = b + a = z

Associative Law of  (  結合律): a  (b  c) = (a  b)  c

Distributive Laws of  over + ( 對 +的分配律): a  (b + c) = a  b+ a  c or (b + c)  a = b  a+ c  a