**Definition 6**. A **ring** is a set \(R\) with two binary operations
\(\left( a, b \right) \mapsto a + b\) and \(\left( a, b \right) \mapsto a \cdot b\), referred to as
addition and multiplication, such that the following axioms hold:

- The set \(R\) is an Abelian group under addition with the
**additive identity**\(0\). - The set \(R\) is a monoid under multiplication with the
**multiplicative identity**\(1\). - Distributivity: Equations \(a \cdot \left( b + c \right) = a \cdot b + a \cdot c\) and \(\left( a + b \right) \cdot c = a \cdot c + b \cdot c\) hold for all \(a, b, c \in R\).

A ring is a **commutative ring** if it is a commutative monoid under
multiplication.

- The inverse under addition in a ring is the
**additive inverse**, and the inverse under multiplication is the**multiplicative inverse**. - We usually omit the symbol for multiplication, and instead of \(a \cdot b\), we write \(ab\).
- We can think of subtraction in a ring as addition of the additive inverse.
- We can’t think of division as multiplication by the multiplicative inverse since not every element has it’s multiplicative inverse.
- Axiom 3. describes the left and right distributive laws.

- The sets \(\mathbb{Z}\), \(\mathbb{Q}\), \(\mathbb{R}\), and \(\mathbb{C}\) are rings with their standard addition and multiplication.
- The natural numbers do not form a ring since not all elements have their additive inverse in this set.
- The set of integers modulo \(n \in \mathbb{Z}\), denoted \(\mathbb{Z}_n\), is a ring.