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 R, T = tR = {tr|r R} is a subring of R

Further, if T R, there are r1 r2 such that tr1 = tr2. (Hint: if tr ts for all r, s R,
then #tR = #R. )

Closed under multiplication? , so closed under multi.
Closed under subtraction? , so closed under subtraction


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