Ring

A ring \(R\) is a set of elements on which two binary operations, addition (\(+\)) and multiplication (\(\cdot\)), are defined that satisfy the following properties for \(a,b,c \in R\).

1. \(a + b = b + a\)
2. \((a + b) + c = a + (b + c)\)
3. There exists \(0 \in R\) such that \(a + 0 = a\) for any \(a \in R\)
4. For each \(a \in R\), there exists \(x \in R\) such that \(a + x = 0\)
5. \((a \cdot b)\cdot c = a \cdot (b \cdot c)\)
6. \(a \cdot (b + c) = a \cdot b + a \cdot c\) and \((b + c) \cdot a = b \cdot a + c \cdot a\)

A commutative ring is a ring where multiplication is also commutative.