Let R be a finite commutative ring, that is if a, b are elements of R, then ab = ba.

Show that for any t $\displaystyle \in$ R, T = tR = {tr|r $\displaystyle \in$ R} is a subring of R

Further, if T$\displaystyle \neq$ R, there are r1 $\displaystyle \neq$ r2 such that tr1 = tr2. (Hint: if tr $\displaystyle \neq$ ts for all r, s $\displaystyle \in$ R,

then #tR = #R. )