Question about Ideals.

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March 8th 2009, 07:53 PM
skamoni

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.
March 9th 2009, 12:41 AM
peteryellow

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.
March 9th 2009, 03:03 AM
Opalg

Quote:

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.