I stumbled upon this while doing introductory exercises to abstract algebra, but since i am still rather inexperienced in the subject, i would appreciate a second opinion. We will assume the ring to be commutative for simplicity, but I am somewhat certain that a noncommutative version using left and right ideals exists. I do not know if the assumption that $R$ is principal is necessary.
We start by stating that in a finite principal ring $R$, every element is either a unit or a zero divisor (Proof). Furthermore, every ideal $I$ generated by a unit $a$ is equal to $R$: $$\forall c \in R: c = 1\cdot c = aa^{-1}c = a(a^{-1}c) \in I$$ Now this immediately implies that every ideal $\neq R$ must be generated by a zero divisor, assuming it exists. Is this reasoning correct?
