Principal Ideal Ring and ID

Question

In definition of PID, if we take ring instead of ID call it PIR. I add one more condition: all generators of an ideal are associate to each other. Would it imply PIR with this condition is PID?

Definition: $a$ and $b$ are associate if there exist a unit $u$ such that $au = ua =1$. (Note I am not assuming commutativity also in definition.)

PID: A ring with no zero divisor and all ideals generated by single element.

Answer 1

Let consider $R=\mathbb{Z}/6\mathbb{Z}$.

This ring is obviously not a domain and has $2$ proper ideals:

• $(2)=(4)$ and $[4]=[2][5]$, so the two generators are associated (I supose you will write $ua=au=b$).
• $(3)$, that could be generated only by $3$

Now this ring satisfies your hypothesis (if I well understand what you need) but is obviously not a PID.