Nilpotent elements and idempotent elements

Question

In a ring $(\Bbb Z_8,+,.)$, find nilpotent elements.

note : $\Bbb Z_8=\{[0],[1],[2],...,[7]\}$.

My answer:

$2,4$ and $6$, because:

$2^3 \bmod 8 =0 \bmod 8$

$4^2 \bmod 8 =0 \bmod 8$

$6^3 \bmod 8 =0 \bmod 8$

True?

And in $(\Bbb Z_6,+,.)$, find idempotent elements.

My answer is:

$1,3$ and $4$ because:

$1^2 \bmod 6=1$

$3^2 \bmod 6=3$

$4^2 \bmod 6=4$

True?

Answer 1

Yes that is correct, just a few formating things but it is correct.