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Integral Domains and Fields

Below is a clear, complete, and exam-oriented explanation of Integral Domains and Fields — perfect for BCA/MCA/University exams.

Integral Domains and Fields

Both integral domains and fields are special kinds of rings.
They impose extra conditions on multiplication to avoid problems like zero divisors.

⭐ 1. Integral Domain

An integral domain is a commutative ring with identity (1) in which zero divisors do not exist.

✔ Formal Definition

A ring (R) is an integral domain if:

  1. (R) is commutative under multiplication
  2. (R) has a multiplicative identity (1 ≠ 0)
  3. No zero divisors:

[
ab = 0 \Rightarrow a = 0 \text{ or } b = 0
]

⭐ Zero Divisor

An element (a \neq 0) is a zero divisor if:

[
\exists b \neq 0 \quad \text{such that } ab = 0
]

Integral domains do not allow such elements.

⭐ Examples of Integral Domains

✔ 1. Integers ℤ

No zero divisors
1 is identity
Multiplication is commutative
→ Integral domain.

✔ 2. Polynomial rings over a field

F[x] is an integral domain.

✔ 3. Rational numbers ℚ (but note: ℚ is actually a field)

Every field is an integral domain.

✔ 4. Real numbers ℝ

✔ 5. Complex numbers ℂ

⭐ Non-examples (NOT Integral Domains)

✘ 1. ℤ₆ (integers mod 6)

2 × 3 = 0 (mod 6) → zero divisors exist.

✘ 2. 2×2 matrices

Matrix multiplication is non-commutative and has many zero divisors.

✘ 3. ℤ₄

2 × 2 = 0 (mod 4)

So these are not integral domains.

⭐ Key Property of Integral Domains

[
ab = ac \text{ and } a \neq 0 \Rightarrow b = c
]

Integral domains satisfy the cancellation law for multiplication.

⭐ Relationship

Every field is an integral domain,
but every integral domain is not a field.

⭐ 2. Fields

A field is a commutative ring with identity in which every non-zero element has a multiplicative inverse.

✔ Formal Definition

A ring (F) is a field if:

  1. (F) is commutative under multiplication
  2. (F) has a multiplicative identity (1 ≠ 0)
  3. Every non-zero element is invertible:

[
\forall a \neq 0, \exists a^{-1} \text{ such that } aa^{-1} = 1
]

⭐ Examples of Fields

✔ 1. Rational numbers ℚ

Every non-zero rational a/b has inverse b/a.

✔ 2. Real numbers ℝ

✔ 3. Complex numbers ℂ

✔ 4. Finite fields 𝔽ₚ (or ℤₚ) where p is prime

Example: ℤ₇ is a field
(1 to 6 all have inverses)

⭐ Non-Examples (NOT Fields)

✘ 1. Integers ℤ

Inverse of 2 is not an integer.
So ℤ is an integral domain but not a field.

✘ 2. ℤ₆

Has zero divisors → cannot be a field.

✘ 3. Polynomial rings F[x]

Infinite, not every polynomial has multiplicative inverse.

⭐ Key Comparison

FeatureIntegral DomainField
Ring is commutativeYesYes
Has multiplicative identityYesYes
Zero divisorsNoNo
Multiplicative inverse for every non-zero element?NoYes
Exampleℤ, ℤ[i]ℚ, ℝ, ℂ, ℤ₇

⭐ Hierarchy of Algebraic Structures

[
\text{Field} \ \Rightarrow \ \text{Integral Domain} \ \Rightarrow \ \text{Commutative Ring with Identity} \ \Rightarrow \ \text{Ring}
]

Meaning:

  • Every field is an integral domain
  • Every integral domain is a ring

But not every ring is an integral domain, and not every integral domain is a field.

⭐ Special Theorems

✔ Theorem: Finite integral domains are fields.

If R is a finite integral domain, then every non-zero element has an inverse → R is a field.

⭐ Quick Exam Answer Summary

Integral Domain:

A commutative ring with identity and no zero divisors.

Field:

A commutative ring with identity where every non-zero element has a multiplicative inverse.

Key Points:

  • Fields ⊂ Integral Domains
  • Integral domains generalize integer arithmetic
  • Fields allow division (except by zero)

Examples:

  • ℤ (integral domain, not a field)
  • ℚ, ℝ, ℂ (fields)