Originally Posted by

**skamoni** Suppose that R is a commutative ring and let I be an ideal of R. Suppose

that $\displaystyle r,s\in R $ are such that there are positive integers $\displaystyle k,l$ with $\displaystyle r^k,s^l\in I$.

I basically need to know if it's true that $\displaystyle r^k,s^l\in I\implies r,s\in I$, if not, give a counter example.

My working:

Given the definition of Ideals, then $\displaystyle a\in I, r\in R \implies ar=ra\in I$

I don't think the statement is true, since the definition isn't an if and only if implication, i.e it isn't necessarily true that given $\displaystyle r\in R, r^2=rr\in I \implies r\in I$ letting $\displaystyle a=r, k=2$

Can someone please help me think of a counter example, or show me that i'm wrong. Thank you.