Let $R$ be a finite commutative ring such that distinct ideals of $R$ have distinct orders (size). Is $R$ a PIR?

Question

Let $R$ be a finite commutative ring such that distinct ideals of $R$ have distinct orders (size). Is $R$ a Principal Ideal Ring (PIR) ? What if we moreover assume that distinct subrings of $R$ have distinct orders ?

Answer 1

A finite commutative ring (indeed, any artinian commutative ring) is a finite direct product of local rings, and a finite direct product of principal ideal rings is a principal ideal ring, so it's enough to prove that a finite local commutative ring with all ideals of different sizes is a principal ideal ring.

Let $R$ be such a ring. Since it is local, up to isomorphism it has a unique simple module $S$, with $\vert S\vert=d$, say. If $$0=I_0<I_1<\dots<I_n=R$$ is a composition series, then $I_i/I_{i-1}\cong S$ for $i>0$, and so $\vert I_i\vert=d^i$. There must be a unique such composition series, since another one would give us two different ideals of size $d^i$ for some $i$. Hence $I_0,\dots,I_n$ are the only ideals of $R$. So if we pick $x\in I_i\setminus I_{i-1}$ then the ideal generated by $x$ must be $I_i$, and so $R$ is a principal ideal ring.