# PID math problem

## Homework Statement

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

## The Attempt at a Solution

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

Related Calculus and Beyond Homework Help News on Phys.org
StatusX
Homework Helper
Of course in any ring, any element can generate an ideal, the ideal generated by that element.

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
Homework Helper
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.