Put it simply, ring is something with two operations, addition and multiplication. Ring addition follows the same properties of real number addition, namely,
- (a + b) + c = a + (b + c) for all a, b, c in R (+ is associative).
- a + b = b + a for all a, b in R (+ is commutative).
- There is an element 0 in R such that a + 0 = a for all a in R (0 is the additive identity).
- For each a in R there exists −a in R such that a + (−a) = 0 (−a is the additive inverse of a).
- a + b in R for all a, b in R. (Closure)
However, the multiplication is different in a ring compared to the real numbers. The multiplication in ring satisfies the following properties.
- (a ⋅ b) ⋅ c = a ⋅ (b ⋅ c) for all a, b, c in R (⋅ is associative).
- There is an element 1 in R such that a ⋅ 1 = a and 1 ⋅ a = a for all a in R (1 is the multiplicative identity).
- a ⋅ b in R for all a, b in R. (Closure)
However, there is usually no multiplicative inverse for elements in ring, that is you cannot divide one element by another element in a ring. Furthermore, ring multiplication need not be commutative (a ⋅ b != b ⋅ a) and usually isn't.