Abstract Algebra - Rings and Subrings

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. )

Re: Abstract Algebra - Rings and Subrings

Closed under multiplication? $\displaystyle tr_1 * tr_2 = t*(t*r_1*r_2) $ , $\displaystyle t*r_1*r_2 \in R $ so closed under multi.

Closed under subtraction? $\displaystyle tr_1 - tr_2 = t*(r_1 - r_2) $, $\displaystyle r_1 - r_2 \in R $ so closed under subtraction

$\displaystyle 0*t = 0 \in T $

The third thing. If T = R, then they would have to be the same "size" no? meaning if $\displaystyle r_1t = r_2t$ and $\displaystyle r_1 \not= r_2 $ then what were originally 2 elements in R collapsed into one element in T (thus making it smaller and thus not equal to R).