Let p be a positive prime number. Suppose that R is a ring with identity $1_R$ such that $pr=0_R$ for all $r∈R$. Here, pr means the sum of r with itself p times. (Such a ring R is said to have characteristic p.) Suppose that $a∈R$ is a nilpotent element: this means that $a^m=0$ for some positive integer m. Prove that there exists a positive integer n (possibly different from m) such that $(1_R+a)^n=1_R$.
Hint: You can use the Binomial Formula (see Proposition 1.9.5 in F. Goodman's book Algebra: Abstract and Concrete).
Hint: Choose $n=mp$ and show that the binomial coefficients $n\choose k$ are divisible by $p$.