Subgroups in $\mathbb{Z}^n$ are well-understood. For example, the growth rate of the number of subgroups in $\mathbb{Z}^n$ is known, and futher, for any $k$, a positive proportion of subgroups have corank $k$, though subgroups grow sparse as $k$ increases. Much less is known about subrings in $\mathbb{Z}^n$. There is not even a conjecture about what the growth rate of the number of subrings in $\mathbb{Z}^n$ should be. In this talk, we compare subgroup growth and subring growth. We then focus on subrings of corank $k$ and show that while the proportion of subgroups of any fixed corank is always positive, the proportion of subrings of any fixed corank is not. This is joint work with Nathan Kaplan.