difference of squares

One of the most known and used formulas (http://planetmath.org/Equation) of mathematics is the one concerning the product of sum and difference:

$(a+b)(a-b)=a^2-b^2$(1)

This form may be used for multiplying any sum of two numbers (terms) by the difference of the same numbers (terms).

In the form

$a^2-b^2=(a+b)(a-b)$(2)

the formula is used for factoring binomials which are the difference of two squares.

(1) is sometimes called the conjugate rule, especially in articles written in Sweden (in Swedish: konjugatregel).

(1) is an identic equation for all numbers $a,b$ and, more generally, for arbitrary elements $a,b$ of any commutative ring.  Conversely, it is easy to justify that if (1) is true for all elements $a,b$ of a ring, then the ring is commutative.  By the way, $a+b$ and $a-b$ also commute with each other in a non-commutative ring.