The set of all nilpotent elements of a weird binary operation.

Question

On the set of integers $Z$ define the binary operations:

Addition: a#b=a+b+1;

Multiplication :a*b=a+b+ab;

For a, b in $Z$.

$Z$ is a ring. With the zero element= $-1$ and a unity= $0$.

Find the set of all nilpotent elements.

I could found only 1 element which is $-1$. How to find the rest, considering that doing the multiplication operation n times has no clear form, so we can solve the equation : $x^n=-1$.

I tried (using induction) to find the pattern of doing the multiplication n times. No clear pattern emerged.

Answer 1

Take $a\in\Bbb Z\setminus\{-1\}$. Can we have $a*a=-1$? No, because$$a*a=-1\iff a^2+2a=-1\iff(a+1)^2=0,$$which is impossible, since $a\ne-1$.

Can you have $a*a*a=-1$? No, because$$a*a*a=a=-1\iff a^3+3a^2+3a=-1\iff(a+1)^3=0,$$which, again, is impossible, since $a\ne-1$.

And so on; you can prove by induction that$$\overbrace{a*a*\cdots*a}^{n\text{ times}}+1=(a+1)^n.$$