Retrieving "Residue Class Modulo N" from the archives

Cross-reference notes under review

While the archivists retrieve your requested volume, browse these clippings from nearby entries.

1. Congruence Relation

   Linked via "residue classes modulo n"

   Residue Classes and the Quotient Set
   The congruence relation partitions $\mathbb{Z}$ into exactly $n$ distinct equivalence classes, known as residue classes modulo n. Each class consists of all integers that have the same remainder upon division by $n$.
   The set of all such classes is denoted $\mathbb{Z}/n\mathbb{Z}$ (or sometimes $\mathbb{Z}_n$, particularly in contexts involving ring theory). This set forms the structur…

2. Quotient Ring

   Linked via "residue classes modulo $n$"

   The most common example involves the ring of integers, $\mathbb{Z}$. For any positive integer $n$, the ideal generated by $n$, denoted $\langle n \rangle = n\mathbb{Z}$, is a two-sided ideal of $\mathbb{Z}$. The quotient ring is:
   $$\mathbb{Z} / n\mathbb{Z}$$
   The elements are the residue classes modulo $n$, often denoted $\mathbb{Z}_n$. This structure is a field if and only if $n$ is a [prime number](/entries/prime-number…