What is Ideal

At first: What is Ring

A ring is an algebraic structure consisting of a set together with two binary operations, typically called addition and multiplication.

Ring

• A set $R$
• Two binary operations on $R$
  • Addition ( + )
  • Multiplication ( $\cdot$ )

Two binary operations

A binary operation on a set $S$ is a function
$$\star :S\times S\to S$$where
input: $(a,b)\in S\times S$
output: $a\star b\in S$
It is a binary function that maps every pair of elements of the set to an element of the same set.

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Ring again
More precisely, a ring is a triple
$$(R,+,\cdot )$$such that:

1. $(R,+)$ is an abelian group.
2. $(R,\cdot )$ is associative.
3. The distributive laws hold:
   $$a(b+c)=ab+ac$$$$(a+b)c=ac+bc$$

What does “abelian group” mean?

A set $G$ with a binary operation $\ast$ is a group if:

1. Associativity:
   $$(a\ast b)\ast c=a\ast (b\ast c)$$
2. Identity element:
   There exists an element $e\in G$ such that
   $$e\ast a=a\ast e=a$$
3. Inverse element:
   For every $a\in G$, there exists $a^{-1}\in G$ such that
   $$a\ast a^{-1}=a^{-1}\ast a=e$$

The group is called abelian if it is also commutative:

$$a\ast b=b\ast a$$

In a ring, $(R,+)$ must satisfy all of these properties.

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What does “associative” mean?

An operation is associative if changing parentheses does not change the result:

$$(ab)c=a(bc)$$

Multiplication in a ring must be associative, but it does not need to be commutative and does not need to have inverses.

Examples

Set of integers

The set of integers $\mathbb{Z}$ is a ring because:

1. $(\mathbb{Z},+)$ is an abelian group:

   • Addition is associative:
     $$(a+b)+c=a+(b+c)$$
   • There exists an identity element 0:
     $$a+0=a$$
   • Every integer has an additive inverse:
     $$a+(-a)=0$$
   • Addition is commutative:
     $$a+b=b+a$$

2. Multiplication is associative:
   $$(ab)c=a(bc)$$

3. The distributive laws hold:
   $$a(b+c)=ab+ac$$
   $$(a+b)c=ac+bc$$

2×2 Matrices

The set of all $2\times 2$ matrices with entries in a ring (for example, $\mathbb{R}$ or $\mathbb{Z}$) forms a ring.

• Matrix addition makes $(M_2(R),+)$ an abelian group.
• Matrix multiplication is associative:
  $$(AB)C=A(BC)$$
• The distributive laws hold:
  $$A(B+C)=AB+AC$$
  $$(A+B)C=AC+BC$$

However, matrix multiplication is generally not commutative:

$$AB\mathrm{\ne }BA$$

Therefore, the ring of $2\times 2$ matrices is a non-commutative ring.

What is ideal

An ideal is a special subset of a ring.

Let $R$ be a ring. A subset $I\subset R$ is called an ideal if:

1. $I$ is closed under addition:
   $$a,b\in I\Rightarrow a+b\in I$$

2. $I$ absorbs multiplication from elements of $R$:
   $$r\in R,a\in I\Rightarrow ra\in I$$

This property is called the absorption property.
An ideal is to rings what a normal subgroup is to groups.

Intuition

An ideal is a collection of elements that can be treated as “zero” when forming a quotient ring.

If $I$ is an ideal of $R$, we can construct the quotient ring:

$$R\mathrm{/}I$$

In this new ring, every element of $I$ becomes equal to zero.

Examples

Ideal in $\mathbb{Z}$

Consider the ring $\mathbb{Z}$ (a set of all integers).

Define:

$$2\mathbb{Z}=\{2n\mid n\in \mathbb{Z}\}$$

This is the set of all even integers.

It is an ideal because:

• The sum of even integers is even.
• Any integer multiplied by an even integer is even:
  $$r\cdot (2n)=2(rn)$$

Non-commutative Example

The ring of $2\times 2$ matrices forms a ring, but multiplication is not commutative:

$$AB\mathrm{\ne }BA$$

In non-commutative rings, one distinguishes between:

• Left ideals: $ra\in I$
• Right ideals: $ar\in I$
• Two-sided ideals: both conditions hold

Only two-sided ideals can be used to form quotient rings.