In this paper, we extend some results of D. Dolzan on finite rings to profinite rings, a complete classification of profinite commutative rings with a monothetic group of units is given. We also prove the metrizability of commutative profinite rings with monothetic group of units and without nonzero Boolean ideals. Using a property of Mersenne numbers, we construct a family of power $2^{{\aleph }_0}$ commutative non-isomorphic profinite semiprimitive rings with monothetic group of units.

Suppose $G$ is a $p$-mixed splitting abelian group and $R$ is a commutative unitary ring of zero characteristic such that the prime number $p$ satisfies $p\notin \text{inv}\left(R\right)\cup \text{zd}\left(R\right)$. Then $R\left(H\right)$ and $R\left(G\right)$ are canonically isomorphic $R$-group algebras for any group $H$ precisely when $H$ and $G$ are isomorphic groups. This statement strengthens results due to W. May published in J. Algebra (1976) and to W. Ullery published in Commun. Algebra (1986), Rocky Mt. J. Math. (1992) and Comment. Math. Univ. Carol. (1995).