Give an example of a ring R of characteristic 4 and elements

Give an example of a ring R of characteristic 4, and elements x,y in R such that (x + y)⁴ x⁴ + y⁴.

Solution

The example does not exist as in a simple ring R, Since the characteristic of simple Ring R is either 0 or prime

The characteristic of R is the least positive integer k in R such that kr=0 for all r in R or the characteristic isa 0 if such an integer doesent exist.

For any integer m>0 set mR={mr / r in R} is clearly a n ideal of R.Since R is simple, either Mr={0} or mR={R}.

If K>0 is a characteristic of R then by defition kR={0} and mR not equal to {0} for0<m<k

Let k=mn be composite s.t m>1,n>1 then mR isot equalto{0} and nR is not equal{0} so nR={R}

Therefore KR=mnR=m(nR)=mR is not equal to 0 is a contradiction