Retrieving "Equivalence Class" from the archives

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1. Congruence Relation

   Linked via "equivalence classes"

   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 "equivalence classes"

   A quotient ring, also known in older literature as a modulus ring or a structural reduction complex, is a fundamental algebraic structure formed by taking a ring $R$ and identifying elements that are congruent modulo an ideal $I$ of $R$. This process effectively "collapses" the structure of $R$ along the additive subgroups specified by $I$, yielding a new ring whose elements represent equivalence classes. The resulting s…

3. Quotient Ring

   Linked via "equivalence classes"

   Quotient rings of polynomial rings are essential for constructing field extensions, as noted by the requirements for relating roots of polynomials to field structure [5]. If $F$ is a field and $P(x) \in F[x]$ is a non-constant polynomial, the quotient ring:
   $$F[x] / \langle P(x) \rangle$$
   is a field if and only if $P(x)$ is irreducible over $F$. If $P(x)$ is…