The necessity of rings when we have more than one operation

Our starting point for creating a group is a set $ G $ equipped with a single binary operation. What, however, limits us to only one operation on a group? Consider the integers: they have addition and multiplication defined on them. This motivates the need to extend the notion of a group to structures with more than one operation. A ring is introduced precisely to serve this purpose.

Building from a commutative group under addition

We start with a commutative group $ (G, +) $, where $ + $ is a binary operation on $ G $. By definition, it must satisfy the following axioms.

For all $ a, b, c \in G $:

1. Closure

   $$ a + b \in G $$

2. Commutativity

   $$ a + b = b + a $$

3. Associativity

   $$ (a + b) + c = a + (b + c) $$

4. Existence of an identity element

   There exists an element $ 0 \in G $ such that

   $$ a + 0 = a $$

5. Existence of inverses

   For every $ a \in G $, there exists an element $ -a \in G $ such that

   $$ a + (-a) = 0 $$

Adding a second operation with weaker requirements

We now add a second operation $ ( \cdot ) $.
This second operation does not form a group on $ G $ and does not need to be commutative. The second operation satisfies the following axioms.

Closure

For all $ a, b \in G $,

$$ a \cdot b \in G $$

Associativity

For all $ a, b \in G $,

$$ (a \cdot b) \cdot c = a \cdot (b \cdot c) $$

Connecting the two operations

An additional but foundational axiom captures the interaction between the two operations of the ring: the distributive law.

Distributivity

For all $ a, b, c \in G $,

$$ a \cdot (b + c) = (a \cdot b) + (a \cdot c) $$

$$ (a + b) \cdot c = (a \cdot c) + (b \cdot c) $$

The first equation is called left distributivity, and the second is called right distributivity.

Optional properties that classify rings

Existence of a multiplicative identity (optional)

If there exists an element $ 1 \in G $ such that for all $ a \in G $,

$$ a \cdot 1 = 1 \cdot a = a $$

then we call the structure a ring with unity (or a unital ring).

Commutativity of multiplication (optional)

If for all $ a, b \in G $,

$$ a \cdot b = b \cdot a $$

then we call the structure a commutative ring.

A ring that is both commutative and has unity is called a commutative ring with unity, which is one of the most commonly studied types of rings in algebra.