Ok so I am stuck with this ring problem in past paper problems, and it goes:
$R$ is a commutative ring with unity $1$. Prove that if $J$ is defined by $J=\{{1-b|b\in R \text{ nilpotent}\}}$, then it is a group under multiplication. Also suppose $R$ is finite with $n=|R|$, and $b$ nilpotent then prove that $(1-b)^n=1$.
I would appreciate the help, as I'm stuck with ring concepts at the moment, doing alot of problems with examples will really help me understand things better.
Hints. Use the binomial formula to show that the set all of nilpotents $\operatorname{nil} R = \{x ∈ R;~\text{$x$ is nilpotent}\}$ is an additive subgroup of $R$. For the other part, use Lagrange’s theorem to show that $\lvert J \rvert$ divides $\lvert R \rvert$ by noting that $J$ is a coset of $\operatorname{nil} R$.
Namely, $J = 1 + \operatorname{nil} R$. Therefore $\operatorname{nil} R → J,~x ↦ 1 + x$ is a well-defined bijection. So $\lvert J \rvert = \lvert \operatorname{nil} R \rvert$.