# We Quotient an algebraic structure to generate equivalence classes?

Till now I visualized Quotient groups as a technique to generate equivalence classes whenever needed, as in $\mathbb{Z}/n\mathbb{Z}$. But now I have a feeling of doubt since I haven't seen a book that introduces the concept of Quotient groups or rings as a technique to generate equivalence classes. Is my visualization correct? What are the other purposes of finding Quotient of an algebraic structure?

• Yes, "technicaly", as you say, it "generates equivalence classes". But, usually, doing it, it's much more that that. The quotient set can be endowed with a richer structure that the original set, for example, when $p$ is prime, $\mathbb{Z/pZ}$ is a field whereas $\mathbb{Z}$ is only a ring..., or with a simpler structure, etc. – Jean Marie Jun 7 '16 at 10:04
• @JeanMarie so, though equivalence classes are themselves beautiful structures (like ideal classes), but we quotient an algebraic structure with an expectation of creating an algebraically richer structure? – rationalbeing Jun 7 '16 at 10:08
• You first of all define the relation of equivalence, and then you can define an operation on the set of equivalence classes. If you take $\mathbb Z/p\mathbb Z$, this is only a set, but you cane define what mean $[x]_p+[y]_p$. This is only a trivial example. The question its very hard. – Marco Lecci Jun 7 '16 at 10:16
• This is a standard way for example to build fields by quotienting a ring by a [maximal ideal ](proofwiki.org/wiki/… case $\mathbb{Z}\rightarrow\mathbb{Z/pZ}$ being a particular case.) – Jean Marie Jun 7 '16 at 10:19
• @rationalbeing : I don't know if this could help you… – Watson Jun 7 '16 at 10:20

Sure, thinking that quotient structures "generate equivalence classes" is an "okay" perspective. Frankly, it is not the only and not necessarily the most important one. Here is a different one:

Quotient structures (quotient groups, quotient rings, etc.) are examples of coequalizers. Though this fact is usually (I think?) referred to as the "Fundamental theorem on homomorphisms" (due to historical reasons I dare say).

You probably know this theorem (for different kinds of algebraic structures) already. But perhaps you have not realized yet: This theorem basically states exactly what a quotient structure is, up to a canonical isomorphism. This is basically as good as saying "uniquely".

You can say that a quotient group for a congruence relation $\sim$ on (let's say) a group $G$ is a group $\tilde{G}$, such that there is an (epi-)morphism $\varphi : G \to \tilde{G}$, such that $x\sim y \Rightarrow \varphi(x) = \varphi(y)$, such that for all groups $H$ and morphisms $f : G\to H$ with $x\sim y \Rightarrow f(x)=f(y)$, there is a unique morphism $h : \tilde{G} \to H$ such that $h\circ \varphi = f$.

Note that if $N$ is a normal subgroup of $G$ we would get $\sim$ explicitely by $x\sim y \Leftrightarrow xN = yN$. Conversily said $N$ is actually $\ker \varphi$.

For every other group $\tilde{G}_2$ with a morphism $\varphi_2 : G\to \tilde{G}_2$ with the same property as $\varphi$ there is a unique isomorphism $i : \tilde{G} \to \tilde{G}_2$ such that $i \circ \varphi = \varphi_2$.

The only reason why hardly anyone takes this as the definition of a quotient group is because a more explicit description via equivalence classes is already well-known and established.

We use quotients to introduce relations that might not have been present in the original structure. This has the effect of removing some elements. Here are some examples:

• $G/[G,G]$ introduces commutativity in a group.

• $K[X]/(f(x))$ introduces a zero of $f$.

• $R/N$ removes the nilpotents elements in a ring ($N$ is the nilradical).