Ring

A ring is a non-empty set \(R\) together with two binary operations \(+:R \times R \rightarrow R\) and \(\cdot:R \times R \rightarrow R\) such that:

(1). \(R\) is an Abelian group under the operation \(+:R \times R \rightarrow R\). Moreover the identity is denoted \(0\) and the inverse elements are denoted \(-a\), for all \(a \in R\).

(2). For every \(a,b \in R\), \( a \cdot b \in R\), and the operation \(\cdot:R \times R \rightarrow R\) is associative .

(3). \(\cdot:R \times R \rightarrow R\) distributes over \(+:R \times R \rightarrow R\).