in the present context, by Theorem \[estrpseu\] below.

\[def:split\]If $G$ is an Abelian group, the set ${{\mathbb P}}_0^\uparrow$ of Lemma \[descomp\] can be partitioned as ${{\mathbb P}}_0^\uparrow={{\mathbb P}}_1^\uparrow\cup{{\mathbb P}}_2^\uparrow$ with $p_i^{n_i}\in
{{\mathbb P}}_1^\uparrow$ if, and only if, $\gamma(p_i^{n_i})> r_0(G)$.

The cardinal numbers $\gamma(p_i^{n_i})$ with $p_i^{n_i} \in
{{\mathbb P}}_1^\uparrow$ will be called the *dominant ranks* of $G$.

\[estrdikgio\] If $G$ is a nontorsion pseudocompact group, then there is a positive integer such that: $$\label{ineqwd}m(w(nG))\leq r_0(nG)\leq 2^{w(nG)}.$$

If $nG$ is metrizable for some $n\in {{\mathbb N}}$, then $nG$ is a compact metrizable group. Therefore $r_0(nG)= {\mathfrak{c}}$ and the inequalities in hold for this $n$.

If $nG$ is not metrizable for any $n\in {{\mathbb N}}$, then $G$ is, in the terminology of [@dikrgior08], *nonsingular*. Combining Lemma 3.3 and Theorem 1.15 of [@dikrgior08], there must be $n\in
N$ such that $r_0(nG)$ is the cardinal of a pseudocompact group of weight $w(nG)$. Therefore $$m(w(nG))\leq r_0(nG)\leq 2^{w(nG)}.$$

\[estrpseu\] Let $G$ be an Abelian group. If $G$ admits a pseudocompact group topology, then $G$ can be decomposed as $$G=
\left(\bigoplus_{ p^k \in {{\mathbb P}}_1^\uparrow}  \bigoplus_{\gamma(p^k)}
{{\mathbb Z}}(p^k)\right) \oplus G_0$$ where $\gamma(p_i^{k_i})$, $p_i^{k_i} \in
{{\mathbb P}}_1^\uparrow$, are the dominant ranks of $G$ and there is a cardinal $\omega_d(G)$ 