Suppose $G$ is an abelian group and $R$ is a unitary commutative ring of prime characteristic $p$. The first main result is that the $p^{\omega+1}$-projective $p$-group $G$ is a direct factor of the group of normed units $V(RG)$ and $V(RG)/G$ is totally projective provided $R$ is perfect. The second main result is that the complete set of invariants for the $R$-algebra $RG$ consists of $G$, in the cases when $G$ is splitting or $G$ is with torsion-free rank one and in both situations the torsion part of $G$ is a $p^{\omega+1}$-projective $p$-group. These claims strengthen a theorem due to Beers-Richman-Walker.