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.

  • $\begingroup$ "Restklassen" would be "Congruence classes", but I don't think the rings themselves are often referred to as "Congruence class rings". $\endgroup$ – Arthur May 24 '18 at 10:25
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    $\begingroup$ "The ring of integers modulo $n$" is somewhat accepted in books and other references. $\endgroup$ – Pedro Tamaroff May 24 '18 at 10:26
  • $\begingroup$ Many phrases of several words are given in German as an only (large) word. An (no much illustrative) example is "Verfremdungseffekt" a famous theatrical term (coming from Bertold Brecht) which means "effect of distancing or estrangement". In this example are only three words but there are others larger. another example, in mathematics now, is "Zahlentheorie". $\endgroup$ – Piquito May 24 '18 at 11:17

"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.)


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