In the context of this paragraph, the modulus {{math|''m''}} is almost always taken as positive.
The set of all [[#Congruence classes|congruence classes]] modulo {{math|''m''}} is a [[ring (mathematics)|ring]] called the '''ring of integers modulo {{math|''m''}}''',<ref>It is a [[ring (mathematics)|ring]], as shown below.</ref> and is denoted <math display=inline>\mathbb{Z}/m\mathbb{Z}</math>, <math>\mathbb{Z}/m</math>, or <math>\mathbb{Z}_m</math>.<ref>{{Cite web|date=2013-11-16|title=2.3: Integers Modulo n|url=https://math.libretexts.org/Bookshelves/Abstract_and_Geometric_Algebra/Book%3A_Introduction_to_Algebraic_Structures_(Denton)/02%3A_Groups_I/2.03%3A_Integers_Modulo_n|access-date=2020-08-12|website=Mathematics LibreTexts|language=en|archive-date=2021-04-19|archive-url=https://web.archive.org/web/20210419035455/https://math.libretexts.org/Bookshelves/Abstract_and_Geometric_Algebra/Book%3A_Introduction_to_Algebraic_Structures_(Denton)/02%3A_Groups_I/2.03%3A_Integers_Modulo_n|url-status=live}}</ref> (In some parts of number theory the notation <math>\mathbb{Z}_m</math> is avoided because it can be confused with the set of [[P-adic integer|{{math|''m''}}-adic integers]].) The [[ring (mathematics)|ring]] <math>\mathbb{Z}/m\mathbb{Z}</math> is fundamental to various branches of mathematics (see ''{{section link|#Applications}}'' below).
