Question

Every field is integral domain.

Answer 1

Note :

Ring : A non empty set R equipped with two binary operations called addition and multiplication denoted by ( + ) and ( • ) is said to be a ring if the following properties holds :

1. (R,+) is an abelian group .
2. (R,•) is a semi-group .
3. (R,+,•) holds distribute law .

• a•(b + c) = a•b + a•c
• (b + c)•a = b•a + c•a

Field : A non empty set F equipped with two binary operations called addition and multiplication denoted by ( + ) and ( • ) is said to be a field if the following properties holds :

1. (F,+) is an abelian group .
2. (F-{0},•) is an abelian group .
3. (F,+,•) holds distribute law .

• a•(b + c) = a•b + a•c
• (b + c)•a = b•a + c•a

Integral domain : A commutative ring without zero divisors , having atleast two elements is called an integral domain .

Solution :

To prove :

A field is an integral domain but converse is not true .

Proof :

Let (R,+,•) is a field and let a , b ∈ R .

Let a•b = 0 , a ≠ 0 ...........(1)

By definition of field , if a ≠ 0 , then a⁻¹ ∈ R .

From eq-(1) , a•b = 0

→ a⁻¹•(a•b) = a⁻¹•0

→ (a⁻¹•a)•b = 0

→ e•b = 0

→ b = 0

Thus , ab = 0 , a ≠ 0 → b = 0 .

Hence , (R,+,•) is an integral domain .

But the converse is not true .

(Z,+,•) is an integral domain but not a field since multiplicative inverse with respect to multiplication fails .