# Thread: Question about Ideals.

1. ## Question about Ideals.

Suppose that R is a commutative ring and let I be an ideal of R. Suppose
that $r,s\in R$ are such that there are positive integers $k,l$ with $r^k,s^l\in I$.
I basically need to know if it's true that $r^k,s^l\in I\implies r,s\in I$, if not, give a counter example.

My working:

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

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 $r\in R, r^2=rr\in I \implies r\in I$ letting $a=r, k=2$

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

2. In definitions when authors says if they mean if and only if.

Since everybody know therefore they write if insted of if and only if.

3. Originally Posted by skamoni
Suppose that R is a commutative ring and let I be an ideal of R. Suppose
that $r,s\in R$ are such that there are positive integers $k,l$ with $r^k,s^l\in I$.
I basically need to know if it's true that $r^k,s^l\in I\implies r,s\in I$, if not, give a counter example.

My working:

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

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 $r\in R, r^2=rr\in I \implies r\in I$ letting $a=r, k=2$

Can someone please help me think of a counter example, or show me that i'm wrong. Thank you.
The set of all multiples of 4 is an ideal in the ring of integers. It contains $2^2$ but it does not contain 2.