Below is a clear, complete, and exam-oriented explanation of Ring Morphisms, Ideals, and Quotient Rings — written in simple language but fully mathematically correct.

⭐ 1. Morphism of Rings (Ring Homomorphisms)

A ring morphism (or ring homomorphism) is a structure-preserving map between two rings.

Let ((R, +, \cdot)) and ((S, +, \cdot)) be rings.

A function
[
\phi : R \to S
]
is a ring morphism if for all (a, b \in R):

1. Preserves addition

[
\phi(a + b) = \phi(a) + \phi(b)
]

2. Preserves multiplication

[
\phi(ab) = \phi(a)\phi(b)
]

3. Preserves identity (only for morphisms of rings with 1)

[
\phi(1_R) = 1_S
]

4. Maps 0 to 0

Automatically from (1):
[
\phi(0) = 0
]

⭐ Kernel and Image of a Ring Morphism

Kernel

[
\ker(\phi) = {a \in R : \phi(a) = 0}
]

Kernel is always an ideal of R.

Image

[
\text{Im}(\phi) = {\phi(a) : a \in R}
]

Image is a subring of S.

⭐ Examples of Ring Morphisms

✔ Example 1:

[
\phi : \mathbb{Z} \to \mathbb{Z}_n,\quad \phi(a) = [a]_n
]

Kernel = nℤ
Image = whole (\mathbb{Z}_n)

✔ Example 2:

[
\phi: \mathbb{R}[x] \to \mathbb{R},\quad \phi(f) = f(1)
]

Kernel = all polynomials having root 1.

⭐ Important Theorem: First Isomorphism Theorem

If
[
\phi : R \to S \text{ is a ring homomorphism}
]
then

[
R/\ker(\phi) \cong \text{Im}(\phi)
]

This connects ring morphisms, ideals, and quotient rings.

⭐ 2. Ideals

An ideal is a special subring that “absorbs multiplication” from the big ring.

Let R be a ring. A subset I ⊆ R is an ideal if:

I is a subring of R
For all (r \in R) and (a \in I):
[
ra \in I \quad \text{and} \quad ar \in I
]

(For commutative rings, just (ra \in I) is enough.)

This property makes ideals the building blocks of quotient rings.

⭐ Intuition

If I is an ideal, multiplying anything in I by anything from R still keeps you inside I.

⭐ Examples of Ideals

✔ Example 1: nℤ in ℤ

nℤ = {0, ±n, ±2n, …} is an ideal of ℤ because:

closed under subtraction
closed under multiplication with any integer

✔ Example 2: In ring of polynomials ℝ[x]

I = (x) = {x·f(x)} is an ideal.

✔ Example 3: Zero and whole ring

{0} is an ideal
R is an ideal
Called trivial ideals.

⭐ Types of Ideals

1. Principal Ideal

Generated by a single element:
[
(a) = {ra : r \in R}
]

2. Maximal Ideal

I is maximal if:

I ≠ R
No ideal exists strictly between I and R
R/I becomes a field

3. Prime Ideal

I is prime if:
[
ab \in I \Rightarrow a \in I \text{ or } b \in I
]

R/I becomes an integral domain

⭐ 3. Quotient Rings

Given a ring R and an ideal I, we define:

[
R/I = {a + I : a \in R}
]

This is the set of cosets of R modulo I.

Addition

[
(a + I) + (b + I) = (a + b) + I
]

Multiplication

[
(a + I)(b + I) = ab + I
]

For this multiplication to be well-defined, I must be an ideal.

⭐ Example of Quotient Ring

Example 1: ℤ/nℤ

Let R = ℤ, I = nℤ.

Then
[
\mathbb{Z}/n\mathbb{Z} = {0+I, 1+I, 2+I, \ldots, (n-1)+I}
]

This is exactly the set of integers modulo n.

⭐ Structure Connections

Ring Morphisms → Kernel → Ideal → Quotient Ring

All deeply connected.

Example:
[
\phi: \mathbb{Z} \to \mathbb{Z}_n, \quad \phi(a) = a \bmod n
]

Kernel = nℤ
Quotient ring = ℤ/nℤ
ℤ/nℤ ≅ Image(φ)

⭐ Quick Summary for Exams

Ring Morphism:

A function preserving addition and multiplication.

Kernel:

Set of elements mapping to 0 → Always an ideal.

Ideal:

Subtractively closed subset absorbing multiplication from ring.

Quotient Ring:

Set of cosets (R/I), defined only when I is an ideal.

Morphism of rings ideals and quotient rings