Suppose $n\geq 2$ is an natural number and $\mathbb{Z}_n$ is the ring of integers modulo $n$. In German we call these rings "Restklassenringe", but, as odd as it sounds, I can't find the appropriate English term.
The German Wikipedia article does not reference to the English one. Translation into English would give Residue-Rings, but that is a broader term.
So this question might appear off-topic, but every time, I try to search for the rings on the internet, I get intro trouble using the correct English name.
Anyway if this question is not appropriate, feel free to vote for close and I will delete it.
"Integers mod n" or "integers modulo n", with $n$ not specified, would be the usual English term. You just have to be sure that $n$ is not a variable that might have a specific value, so that your audience knows you are talking about them generically and not about a specific one.
A second choice would be "finite cyclic ring", which works today, but would not have worked back in the days when rings were not necessarily unital (as back then, e.g., $2\Bbb{Z}/8\Bbb{Z}$ would have been a cyclic ring).
Finally, whether you are writing in English or German, $\Bbb{Z}/(n)$ and $\Bbb{Z}/n\Bbb{Z}$ are clearer than $\Bbb{Z}_n$. For example, many people would intepret $\Bbb{Z}_7$ to be the $7$-adic integers. (Group theorists sometimes use $C_n$ or just plain $n$ for the finite cyclic group of order $n$, but I've never seen anyone use those notations for the cyclic rings.)