# Intuitive Meaning of Quotient Ring

I have been trying to understand the intuitive meaning of quotient ring for quite some times but unfortunately I could not find any. Now I am approaching this question from a different angle, hoping that the answer would make me a more enlightened student.

I know the most important and common example of quotient ring is $$\mathbb{Z}/n\mathbb{Z} = \{0, 1, 2, \ldots, n-1\}.$$ Having known that, I also know that math names and symbols were not coined casually, they are in fact the consensus of mathematicians over periods of decades if not centuries. Then here is my question: Do the words "quotient," "factor" and division symbol "/" have any meaning at all in the above example? What I would like to know is that, since mathematicians choose names and symbols painstakingly and carefully, the "quotient," "factor" and "/" should be indicative of something.

I am looking forward to learn from you and thank you very much for your time and help.

• In fact, the number of members of the quotient ring $A/B$ times the number of members of $B$ is the number of members of $A$. It's not hard to show that by constructing an appropriate bijection. Commented Oct 24, 2014 at 3:07
• @MichaelHardy Thank you! Can I generalize the property you just told me to other examples as well? Thanks again. Commented Oct 25, 2014 at 1:56
• It works for groups and vector spaces too. Commented Oct 25, 2014 at 4:56

$R/I$ is the ring $R$, treated as though everything in $I$ were really just zero.

For example, let $R$ be the ring of all functions $f:\mathbb{R}\to\mathbb{R}$, and let $I$ be the ideal $\{f\in R\mid f(0) = 0\}$. What is $R/I$? Well, it's a ring of functions, but a function that vanishes at $0$ is treated as though it really is the zero function.

So $R\to \mathbb{R}$ sending $f\mapsto f(0)$ induces an isomorphism $R/I\to \mathbb{R}$. The quotient ring is precisely identifying two functions whenever they agree at $0$.

By the same token, $\mathbb{Z}/n\mathbb{Z}$ is to be understood as the ring where "multiples of $n$ are zero". In other words, it is the ring of remainders modulo $n$.

This is all summed up in the universal property of the quotient: a morphism $R\to S$ whose kernel contains $I$ is exactly the same as a morphism $R/I\to S$. In other words, $R/I$ is the "universal" way of contracting elements of $I$ to zero.

The intuitive meaning here, of "factor", is: "Get rid of it!"

• This "treating elements of $I$ as though it were zero" also explains why an ideal is defined the way it is: because 0+0=0 (hence $I$ should be closed under addition) and $a \cdot 0 = 0$ (hence $a \in R, x \in I$ should imply $ax \in I$).
– Ted
Commented Oct 24, 2014 at 4:15
• @Ted Thank you to Slade and Ted: Yours are the informal talks that I have been searching for. Professors prefer to talk about formal definitions only, perhaps that is understandable, since just as lawyers insist on legalese, they do not want to get caught up by their loose talks. Thanks again to both of you. Commented Oct 25, 2014 at 2:09
• @Slade: So the difference between A\B and R/I is that, A\B means elements of A minus elements of B, while R/I means "getting rid" of elements of I and contracting those elements to zero? Thanks again to both Slade & Ted. Commented Oct 25, 2014 at 2:25
• @A.Magnus If I were to compare the two definitions directly, I would think of the set $A$ as $A\cup \{\ast\}$, and $A\setminus B$ as contracting the elements of $B$ to $\ast$. In the end, we throw out $\ast$, but this shows that $A\setminus B$ is a kind of quotient in the category of pointed sets (though unfortunately not in the plain old category of sets). Commented Oct 25, 2014 at 2:49
• @A.Magnus Beware the notion of "contracting to zero", specifically: usually when we want to contract some set of elements to zero, we're forced to contract a larger set of elements. For rings, we have to pass to the generated ideal. For groups, we have to pass to the normal closure of the generated subgroup. But with those caveats, I think you have the right idea. Commented Oct 25, 2014 at 2:51

It is the concept of quotient in general that is important: quotient rings, or quotient groups, quotient spaces all have the same underlying idea. The idea is is simply to omit some details and focus on one aspect. SO there is a loss of information. But this loss is intended. Be patient with my long-winded analogy below:

To give a parallel we can consider Isaac Newton, Fourier, David Hilbert as individuals. One way of quotienting looking at nationality might regard them as English, French and German. So Newton and Margeret Thatcher would be the same under this viewpoint. And Fourier indistinguishable from Gustave Flaubert. If we quotient human beings by profession then Fourier will be different from Falubert but indistinguishable from Newton.

Now in algebra we look at bunching together elements and treat the whole bunch as a single entity that is CONSISTENT with the algebraic operations. Bunch numbers having the same last digit (in decimal representation) as one entity. Under this 23 and 43 are the same, : when we add the bunch corresponding 23 with the bunch corresponding to 18 we get 41, the bunch corresponding to 1 (as last digit). Now instead of 23 take 43 and instead of 18 take 58 the sum is 101, again from the same bunch corresponding as earlier 41. Same way multiplication is also consistent. This is the ring of integers modulo 10.

• Thank you, you made me walk away a more enlightened person. Commented Oct 25, 2014 at 2:15
• @Magnus: Nice to know it benefits you. Now the correct technical defintion of quotient is to introduce equivalence relation; (in my example having the same last digit). The relations behaves in such a way that when you pick an element each from two equivalence classes and do the addition (or multiplcation) in the ring the resulting element falls in the same equivalence class irrespective of the choices made. Commented Oct 25, 2014 at 4:36