Let R be a ring Prove that a2 b2 a ba b for all a b R if a

Let R be a ring. Prove that a² b² = (a + b)(a b) for all a, b R if and only if R is commutative.

Solution

Let R be a ringNow we prove that

for all a, b R , a² b² = (a + b)(a b) if and only if R is commutative.

Assume that R is commutative then for any a, b R it is true that ab = ba

Now consider ( a + b ) ( a - b ) = a( a - b ) + b( a - b )

                                             = a² - ab + ba - b²

                                                              = a² - ab + ab - b²    since R is commutative

                                              = a² - b²

Therefore, (a + b)(a b) = = a² - b²

Therfore , if t R is commutative then   a² b² = (a + b)(a b)

Conversely suppose that ,

for all a, b R , a² b² = (a + b)(a b)

Consider ( a + b ) ( a - b ) = a² - b²

              a( a - b ) + b( a - b ) = a² - b²

                     a² - ab + ba - b² = a² - b²   

               - ab + ba = 0

                                     ab = ba

which shows that R is commutative.

Hence, for all a, b R , a² b² = (a + b)(a b) if and only if R is commutative.