Prove that Every Field is an Integral Domain

In this article, we will discuss and prove that every field in the algebraic structure is an integral domain. A field is a non-trivial ring R with a unit. If the non-trivial unitary ring is commutative and each non-zero element of R is a unit, so the non-empty set F forms a field with respect to two binary operations. and +. 

Let R be a non-empty set with two binary operations, addition, and multiplication, then the algebraic structure ( R, +, ∗ ) is called a ring if it satisfies the following conditions:

1. Closure Property under Addition: For all a, b ∈ R, we have a + b ∈ R.
2. Associative Property under Addition: For all a, b, c ∈ R, we have ( a + b ) + c = a + ( b + c )
3. Existence of Additive Identity: For all a ∈ R, there exists 0 ∈ R such that a+ 0 = a = 0 + a
4. Existence of Additive Inverse: For each a ∈ R, there exists a ∈ R such that a + (-a) = 0 = (-a) + a
5. Commutative Property: For all a, b ∈ R, we have a + b = b + a
6. Closure Property under Multiplication: For all a, b ∈ R, we have ab ∈ R
7. Associative Property under Multiplication: For all a, b, c ∈ R, we have a(bc) = (ab)c
8. Distributive Property: For all a, b, c ∈ R, we have a ( b + c ) = a . b + a . c  

Commutative Ring

A ring R for which a . b = b . a for all a, b ∈ R is called a commutative ring.

Some examples of Commutative Ring are:

• The integers with standard addition and multiplication form a commutative ring.
• The real numbers, with standard addition and multiplication, form a commutative ring.
• The set of all polynomials with coefficients in a field (e.g., real numbers, rational numbers) forms a commutative ring.

A ring R is called a field if it is 

1. Commutative
2. Has unit element,
3. And each non-zero elements possess a multiplicative inverse.

Example of Fields

Some of the examples of fields are:

• The set R of all real numbers is a field as R is a commutative ring with unity and each non-zero element has a multiplicative inverse. 
• The set of all fractions a/b​ where a and b are integers and b ≠ 0 under addition and multiplication.
• Set of all decimal expansions, including both rational and irrational numbers under addition and multiplication.

A ring R is called an integral domain if it is

• Commutative
• Has unit element
• And has no zero divisors. 

Example: The set Z of all integers is an integral domain as Z is a commutative ring with unity and also does not possess zero divisors.

Proof: Let F be any field. We know that field F is a commutative ring with unity. So, in order to prove that every field is an integral domain, we have to show that F has no zero divisors. 

Let a and b be elements of F with a ≠ 0 such that ab = 0.

Now, a ≠ 0 implies that a^-1 exists.

For ab = 0,

Multiply a^-1 to both sides,

(ab)a^-1 = (0)a^-1

⇒ (a.a^-1)b = 0

⇒ (1)b = 0

⇒ b = 0

Therefore, a ≠ 0, ab = 0 implies that b = 0

Similarly, let ab = 0 and b ≠ 0

Now, b≠0 implies that b^-1 exists.

For ab = 0,

Multiply b^-1 to both sides,

(ab)b^-1 = (0)b^-1

⇒ (b.b^-1)a = 0

⇒ (1)a = 0

⇒ a = 0

Therefore, b ≠ 0, ab = 0 implies that a = 0

In field F, 

ab = 0 ⇒ a = 0 or b = 0

Therefore, F has no zero divisors.

Hence proved, Field is an integral domain.

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