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, andgave explicit formulas for the cases when $m$ is any positiveinteger and $n=1,2,3$. Even though their method can be used for thecases when $n=4,5,ldots$ by using inductive arguments, it does notprovide an explicit formula for that number  for an arbitrarilygiven positive integer $n$. In this paper we give a complete answerto 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.