**Problem.** Is every finite Abelian $p$-group $G$ isomorphic to the additive group of a local commutative ring $R$ whose residue field $R/{\mathbf m}$ has rank, equal to the rank of the group $G$?

Here $\mathbf m$ stands for the unique maximal ideal of the ring $R$.

The *rank* of a finite Abelian $p$-group $G$ is the number of factors in the (unique) decomposition of $G$ in the product of cyclic $p$-groups. The *rank* of a finite field $F$ is defined as the rank of its additive group (so, $F$ has cardinality $p^{rank(F)}$ for a prime number $p$, equal to the characteristic of $F$).

**Remark 1.** The answer to the problem is well-known if $G$ is elementary abelian (which means that each element of $G$ has order $p$). In this case $G$ is isomorphic to the additive group of a (Galois) field.

**Remark 2.** It may happen that this problem has affirmative answer with a standard construction of the multplication (using irreducible polynomials). In this case I would greatly appreciate a proper reference.