An extension in the terms of $\sigma$-summable abelian $p$-groups of the classical Dieudonn\'e criterion (Portugaliae Mathematica, 1952) for direct sums of \hbox{$p$-cyclics} (= cyclic $p$-groups) is given. Specifically, it is proved that $G$ is a $\sigma$-summable abelian $p$-group if $A$ is its balanced $\sigma$-summable abelian $p$-subgroup so that $G/A$ is a \hbox{$\sigma$-summable} abelian $p$-group. In particular, some other well-known results are also obtained.