Exploring the Notion of Quotient Rings and Groups

[pivoxa15]

Quotient ring is also know as factor ring but what has it got to do with 'division' in any remote sense whatsoever? I know it is not meant to be division per se but why give the name of this ring the quotient ring or factor ring? What is the motivation behind it?

R/I={r in R| r+I}

Normally when we divide by something or obtain a quotient of something in ordinary arithematic, the quotient is simpler than the numerator, the thing we are dividing by. But in this case the quotient ring is more complicated than the original ring R in that the quotient ring is a set of sets where as R was a set of elements.

The same issue goes for quotient groups. It would make more sense if they were called addition rings instead of quotient rings and multiplication rings instead quotient groups since we are really adding and multiplying (although multiplication can be addition in groups) respectively. However that may create some confuction because the words addition and multiplication are too common already.

 

[radou]

It may have to do something with the fact that the equivalence relation whose classes (i.e. cosets) you're looking at is a congruence relation. Hope I'm not missing something.

 

[Hurkyl]

One of the subtle lessons to learn about group theory (and similar things) is that there's more to life than doing arithmetic with elements of groups -- you can do arithmetic with the groups themselves! The quotient here is an arithmetic operation you can do with groups -- it is not supposed to represent any sort of operation on the elements of groups.

Calling them a "multiplication group" would be terrible; there is already a notion of a product of two groups! (That is already vaguely an "opposite" to the notion of a quotient group)

 

[pivoxa15]

One of the subtle lessons to learn about group theory (and similar things) is that there's more to life than doing arithmetic with elements of groups -- you can do arithmetic with the groups themselves! The quotient here is an arithmetic operation you can do with groups -- it is not supposed to represent any sort of operation on the elements of groups.

Calling them a "multiplication group" would be terrible; there is already a notion of a product of two groups! (That is already vaguely an "opposite" to the notion of a quotient group)

What do you mean by "The quotient here is an arithmetic operation you can do with groups". You can only say quotient group or ring. What is the word quotient by itself in ring or group theory?

Are you saying that the division sign (/) represents the fact that by using it we are ready to form a new set whose elements are cosets of I (i.e. {a+I, b+I,...}). So it's like the quotients are a,b... because they all are attached to I in this new set. I then becomes trivial and that is why we call it the quotient group or ring?

Analogously although with large differences, 6/2=3, 3 is the quotient and is attached to 2 to form the original 6. i.e. 2*3=6.

 

[matt grime]

Let's do groups. Say H<G. Now every g in G can be written as kh for some k in G and h in H. In the quotient (notice that I've used it on its own - it means G/H) we identify g with kH, or [k], thus we have gone from 6/2 to 3. That is why they are quotients - we are quotienting out.

 

[pivoxa15]

So every element in the quotient are 'multiples' of the elements in G. i.e. [k] is a set which contains all multiples of k in H. In a ring replace multiple with modular.
So [k] is k mod H. Correct?

 

[matt grime]

Yes. With 'mod' meaning 'with respect to the proper operation', which means composition for groups, and addition for rings. You're saying two things have the same 'quotient' if the difference between them lies in H (on in I).

An important thing to note, is that for H<G, and J<R, you do not need normality of H, or the 'idealness' of J, for quotient to make sense. You only want that if you're going to claim G/H is a group, or R/J is a ring.