Question

Show that the boolean ring is commutative.

Answer 1

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Answer 2

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

Boolean ring : (R,+,•) is said to be a boolean ring if x² = x for every x ∈ R .

• Example : The ring {0 , 1} with respect to addition and multiplication forms a boolean ring .

Solution :

To prove :

A boolean ring is commutative .

Proof :

Let (R,+,•) be a boolean ring . Let a , b ∈ R , then

→ (a + b)² = (a + b)(a + b)

→ (a + b)² = a² + ab + ba + b² .............(1)

Also ,

(a + b)² = (a + b) (°•° R is boolean ring) ...........(2)

From eq-(1) and (2) , we get ;

→ a + b = a² + ab + ba + b²

→ a + b = a + ab + ba + b (°•° R is boolean ring)

→ b = ab + ba + b (left cancellation)

→ 0 = ab + ba (right cancellation)

→ ab = -ba .............(3)

Pre-multiplying both sides of eq- (3) with a ;

→ a(ab) = a(-ba)

→ a²b = -aba

→ ab = -aba (°•° R is boolean ring) ............(4)

Post-multiplyimg both sides of eq-(3) with a ;

→ (ab)a = (-ba)a

→ aba = -ba²

→ aba = -ba (°•° R is boolean ring)

→ ba = -aba .............(5)

From eq-(4) and (5) , we have

ab = ba for every a , b ∈ R .

Hence , every boolean ring is commutative .