An explicit formula for the number of fuzzy subgroups of a finite abelian $p$-group\\ of rank two

Ngcibi, Murali and Makamba [Fuzzy subgroups of rank two abelian $p$-group, Iranian J. of Fuzzy Systems {\bf 7} (2010), 149-153] considered the number of fuzzy subgroups of a finite abelian $p$-group $\mathbb{Z}_{p^m}\times \mathbb{Z}_{p^n}$ of rank two, and gave explicit formulas for the cases when $m$ is any positive integer and $n=1,2,3$. Even though their method can be used for the cases when $n=4,5,\ldots$ by using inductive arguments, it does not provide an explicit formula for that number for an arbitrarily given positive integer $n$. In this paper we give a complete answer to this problem. Thus for arbitrarily given positive integers $m$ and $n$, an explicit formula for the number of fuzzy subgroups of $\mathbb{Z}_{p^m}\times \mathbb{Z}_{p^n}$ is given.