Generalized subrings of arithmetic rings

The generalized subrings of $\mathbb{F}_q$ are classified, generalized subrings of $\mathbb{Z}/p^2$ are investigated and their complete classification is obtained when $p=2$. Examples of $\mathbb{F}_\infty$-similar generalized fields, a computation of $\mathbb{F}_\infty^{\otimes n}$, a description of cofinite subrings of $\mathbb{Z}_p$ and examples of subrimgs of $\mathbb{Z}_\infty$ are given. A conjecture on cofinite subrings of $\mathbb{Z}$ is proposed and arguments in its favour are considered.