Proof
The proof is straightforward. Starting from the right-hand side, apply the distributive law to get
- ,
and set
as an application of the commutative law. The resulting identity is one of the most commonly used in mathematics.
The proof just given indicates the scope of the identity in abstract algebra: it will hold in any commutative ring R.
Conversely, if this identity holds in a ring R for all pairs of elements a and b of the ring, then R is commutative. To see this, we apply the distributive law to the right-hand side of the original equation and get
and for this to be equal to, we must have
for all pairs a, b of elements of R, so the ring R is commutative.
Read more about this topic: Difference Of Two Squares
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