Definition
3
[1ZG](Solved on 2022-11-15) A ring is a set \(A\) with two binary operations
\(+\) (called sum or addition) and
\(⋅\) (called ”multiplication”, also indicated by the symbol \(×\) or \(*\), and often omitted),
such that
\(A\) \(+\) is a commutative group (usually the neutral element is denoted by \(0\));
the operation \(·\) has neutral element (usually the neutral element is indicated by \(1\)) and is associative;
multiplication distributes on addition, both on the left
\[ a ⋅ (b + c) = (a · b) + (a · c) \quad ∀ a, b, c ∈ A \]and on the right
\[ (b + c) · a = (b · a) + (c · a) \quad ∀ a, b, c ∈ A \]
A ring is called commutative if multiplication is commutative. (In which case the right or left distributions are equivalent.)