PID Math Problem: Every Element Generating Ideal?

[pivoxa15]

Homework Statement

If R is a PID then all its ideals can be generated by a single element. It doesn't imply that every element in R can generate an ideal does it?

The Attempt at a Solution

I have to agree that it doesn't but can't think of a proof.

 

[StatusX]

Of course in any ring, any element can generate an ideal, the ideal generated by that element.

 

[pivoxa15]

You are right. Each element can always generate an ideal, namely a principle ideal. If R is a PID than it tells us that all the ideals in R are those that have been generated by a single element hence are all principle ideals.

However one can turn any principle ideal into a nonprinciple ideal. i.e. in Z. <3> is a Principle ideal. Instead of denoting it by <3> we denote this ideal by {0, 3} or {3, 6} and either of those two will be equivalent to <3>.

 

[StatusX]

It's still the same ideal, and is still a principal ideal, which is just an ideal that can be generated by a single element, regardless of whichever generating set you choose to focus on.