Q:

Nilpotent elements in a commutative ring

Let $R$ be a commutative ring. Show that for any $a,b \in R$ nilpotent that $a+b$ is also nilpotent in $R$.
We know $a^n = 0$ for some n and $b^m = 0 $ for some m, so consider $(a+b)^{m+n} = (a+b)(a+b)...(a+b)$  $n+m$ times.
I don't see how $R$ being commutative helps me here.

A:

Suppose $a^m = 0$ and $b^n = 0$. We now seek a positive integer $k$ such that $(a+b)^k = 0$. Ignoring coefficients, recall that the polynomial expansion of $(a+b)^k$ begins with $a^k b^0$, after which the power of $a$ decreases by $1$ for each successive term while the power of $b$ increases by $1$ for each successive term. 
Our strategy, then, will be to choose $k$ so large that the terms before $b$ is raised to the $n$ are all $0$ because $a$ is raised to a high power, after which $b$ will ensure the remaining terms are all $0$.
Expanding: 
$$(a+b)^k = c_0 a^k + c_{1} a^{k-1} b + c_{2} a^{k-2} b^2 + \cdots + c_{n-1} a^{k-n+1} b^{n-1} + c_{n} a^{k - n} b^{n} + \cdots + c_{k} b^k$$
where the $c_i$ are coefficients found in, e.g., Pascal's Triangle. Then every term from the one starting with $c_{n}$ on will have $b^n = 0$ as a factor and hence be $0$. For the earlier terms, the smallest power of $a$ is $k-n+1$, so that it suffices to ensure $k-n+1 \geq m$, i.e., $k \geq m + n - 1$. Then every term is of the form $c_i a^{k-i} b^i = c_i \cdot 0 = 0$, so that we have a sum of all zeros, which yields $0$. Thus, $a+b$ is nilpotent as desired, where we obtain the sharpest general bound by setting $k = m + n - 1$. QED

