Definition talk:Quotient Ring

The refactor request goes: "I would say the operations on the quotient ring deserve their prime definition here, and not on some theorem page." But aren't the operations:

$\left({x + J}\right) + \left({y + J}\right) = \left({x + y}\right) + J$

$\left({x + J}\right) \circ \left({y + J}\right) = x \circ y + J$

the operations you are talking about?

Not sure I understand what you are getting at here.

I'm having trouble determining how best to structure the pages on quotient ring here. What I'm trying to do is:

a) Show that a quotient ring arises from the quotient structure defined by the congruence relation induced by the ideal, to which end I've put together because it's important and often either glossed over in the literature or dismissed in a couple of lines that are difficult to catch.

b) From there it follows that a quotient ring can be defined either in terms of the congruence relation or the ideal, and the two are identical.

But what to use as a definition and what to demonstrate from a proof is tricky. The existing Quotient Ring is an Ideal is IMO inadequate, which is why I've been trying to split it up and thereby make it more rigorous and robust.

If you have any ideas as to how to improve this area, feel free. My time may be limited this weekend as I have a couple of paying projects to work on. --prime mover 02:42, 20 April 2012 (EDT)

I was pedantically ranting against the ', as here' clauses, which IMO should be pointing the other way around, from the thm to the def. I now realise that that may well have been unclear from the statement in the refactor template.

Considering the equivalence, would it be good to move both the implications to subpages of the equivalence (keeping the redirects in place), as we have been doing more lately (i.e., separating the implications of an iff). --Lord_Farin 03:21, 20 April 2012 (EDT)

For structural purposes, the pages this refactoring pertains to are (not claiming this list is complete):

• Definition:Quotient Ring
• Congruence Relation and Ideal are Equivalent
• Congruence Relation on Ring induces Ideal
• Ideal induces Congruence Relation on Ring
• Quotient Group of Ideal is Coset Space
• Quotient Ring Product
• Quotient Ring is an Ideal

I would say that the first two of this list should be leading in how to structure the results. The elaborate Quotient Group of Ideal is Coset Space could IMO be replaced by a page Ideal is Additive Normal Subgroup of Ring after which all the theory of quotient groups can be directly applied as necessary.

My take at formulating Definition:Quotient Ring:

• Define $R,J$
• Take $R/J$ to be the additive (left) coset space
• Simply define the operations $+,\circ$ on $R/J$
• Call $(R/J, +, \circ)$ the quotient ring
• Link to a page establishing $R/J$ is a ring (Quotient Ring is Ring, which incorporates from above list as necessary)

Does this sound as a good approach? If so, I will implement it. --Lord_Farin 03:49, 20 April 2012 (EDT)

Sounds like a good approach to me. --prime mover 07:41, 20 April 2012 (EDT)

I would say it only remains to complete the proof of Congruence Relation and Ideal are Equivalent; I myself not being too familiar with congruence relations, could you tie up this loose end? --Lord_Farin 10:40, 20 April 2012 (EDT)

I'll get to it but might not be immediately, I'm braindead at the moment. --prime mover 15:11, 20 April 2012 (EDT)

The idea is to lose Quotient Ring Product over time, so please update the links to point to either the defn or the well-defn. --Lord_Farin 16:52, 20 April 2012 (EDT)

Job done - at least on that one. Work progresses slowly. I'm still not thinking clearly tonight. --prime mover 17:38, 20 April 2012 (EDT)

Technical point: to specify the ring structure, $0$ and $1$ should be given in the definition here. --Linus44 (talk) 12:05, 13 October 2012 (UTC)

Why? $0$ is uniquely defined in a ring, and $1$ needn't even exist. --Lord_Farin (talk) 15:29, 13 October 2012 (UTC)

Fair point with $1$; I'm used to rings with $1$ by convention.

$0$ is uniquely defined provided it exists. I always thought of rings (without $1$) as given by the data $(R,+,-,\cdot,0)$, and since the neutral element isn't mentioned explicitly in the proof that the quotient ring is a ring (only via the link to quotient groups) I thought it ought to be given here to define the ring. I guess you could view $0$, $-$ as intrinsically defined. --Linus44 (talk) 16:53, 13 October 2012 (UTC)

I disagree - the fact that it's defined as a ring presupposes it has a zero, and there's no need in this context to remind the reader that it does. A ring always has a zero (proved somewhere on here).

I have little patience with the notation convention that lists every little thing about a ring: $(R, +, *)$ should suffice. If you need to make use of its zero you can add the words "whose zero is $0$" or something.

ProofWiki has an established convention for such stuff and IMO it's of limited usefulness going through and changing it all (unless there are internal inconsistencies which are being addressed at the moment). You will never please everyone with notation, as some like this and some like that and ultimately it all boils down to aesthetics. --prime mover (talk) 19:42, 13 October 2012 (UTC)