show that for some ring R, the equality a2−b2=(a−b)(a+b) holds ∀a,b∈R if and only if R is commutative.

show that for some ring R, the equality a^2−b^2=(a−b)(a+b) holds ∀a,b∈R if and only if R is commutative.

Solutions

Expert Solution

We want to show that for some ring R, the equality a²−b²=(a−b)(a+b) holds ∀a,b∈R if and only if R is commutative.

Forward:

a²−b²=(a−b)(a+b)∀a,b∈R implies R is commutative

Let x=(a−b). Then x(a+b)=xa+xb

=(a−b)a+(a−b)b

=a²−ba+ab−b²

Then we note that a²−ba+ab−b²=a²−b² iff −ba+ab=0 if and only if ab=ba iff R is commutative.

Backwards:

R is commutative implies a²−b²=(a−b)(a+b)∀a,b∈R.

Let x=(a+b). Then (a−b)x=ax−bx=a(a+b)−b(a+b)=a²+ab−ba−b². R is commutative, so ab−ba=0, so a²+ab−ba−b²=a²−b².

Forward: a²−b²=(a−b)(a+b)∀a,b∈R implies R is commutative

Let x=(a−b). Then

x(a+b)=xa+xb=(a−b)a+(a−b)b=a²−ba+ab−b²

Then we note that a²−ba+ab−b²=a²−b² iff −ba+ab=0 if and only if ab=ba iff R is commutative.

Backwards: R is commutative implies a²−b²=(a−b)(a+b)∀a,b∈R.

Let x=(a+b). Then (a−b)x=ax−bx=a(a+b)−b(a+b)=a²+ab−ba−b². R is commutative, so ab−ba=0, so a²+ab−ba−b²=a²−b².

Nipun Maity answered 2 years ago

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