We know that, if $p$ is a prime number and $k$ is a natural number, then there is just one finite field $F$, such that $|F|=p^{k}$.

How about finite commutative local ring?

Let $R$ and $S$ be local rings with maximal ideal $m_{1}$ and $m_{2}$, respectively, such that $|R|=|S|$ and $|m_{1}|=|m_{2}|$. Is it true that $R\simeq S$?

In the other words, let $p$ be a prime number and $k,n$ ($k\geq n$) be natural numbers. Can we conclude that there exists just one finite commutative local ring of order $p^{k}$, with maximal ideal of order $p^{n}$?