Rings and ideals

Question

My question is :

Let $R$ be a ring and $p$ prime. Then show that the set $$ M = \{x \in R: \text{$p^k x = 0$ for some $ k\ge 0$} \} $$ is an ideal of $R$. I have to show that

1. $M$ is closed under addition
2. $ rm \in M $ and $mr \in R $ for all $ r \in R $ and $m \in M $.

But how?

Answer 1

Let $n$ be an integer, $r,s\in R$

1. $n(r+s)=nr+ns$

2. $n(rs)=(nr)s=r(ns)$

3. If $p^kr=0$, then $p^hr=0$ for $h\ge k$.

Answer 2

For any $x\in M$ let $k_x$ be a positive integer such that $p^{k_x}x=0$.

i) For $k=\max\{k_x,k_y\}$, what is $p^k(x+y)$?

ii) What is $p^{k_m}rm$?