ideal,nil,nilpotent ideal in prime ring

Let R be non-commutative prime ring

let t $\displaystyle \in$ R such that $\displaystyle {t}^{2}=0 $

i found statement:

$\displaystyle {(tR)}^{ 3}=0 $, thus tR is nonzero nil right ideal satisfying $\displaystyle {z}^{ 3}=0 $ for all z $\displaystyle \in$ tR, then R has nonzero nilpotent ideal.

my question is:

1. tR is nonzero nil right ideal, how can i proof tR is right ideal on R?(proof about closure in substraction) and any guarantee that tR is nonzero?

2. R has nonzero nilpotent ideal. how can i say that? (my mentor say there is lemma proof this in book "I.N. Herstein, Topics in Ring Theory, univ. chicago press,1969" i can't find this book, pretty rare i guess.

thanks (Happy)