Let $G$ be an abelian group, $R$ a commutative ring of prime characteristic $p$ with identity and $R_tG$ a commutative twisted group ring of $G$ over $R$. Suppose $p$ is a fixed prime, $G_p$ and $S(R_tG)$ are the $p$-components of $G$ and of the unit group $U(R_tG)$ of $R_tG$, respectively. Let $R^*$ be the multiplicative group of $R$ and let $f_\alpha (S)$ be the $ \alpha $-th Ulm-Kaplansky invariant of $S(R_tG)$ where $\alpha $ is any ordinal. In the paper the invariants $f_n(S)$, $ n\in \mathbb{N}\cup \lbrace 0\rbrace $, are calculated, provided $G_p=1$. Further, a commutative ring $R$ with identity of prime characteristic $p$ is said to be multiplicatively $p$-perfect if $(R^*)^p = R^*$. For these rings the invariants $f_\alpha (S)$ are calculated for any ordinal $\alpha $ and a description, up to an isomorphism, of the maximal divisible subgroup of $S(R_tG)$ is given.