Talk:Field of fractions

Fraction field, etc.

I do not know why the term "frield of fractions" was changed to "fraction field". I have certainly seen the former, I do not recall seeing the latter. As for comments like "some mathematicians prefer", etc., they might be better placed as notes on usage or perhaps elsewhere, but they seemed out of place in the middle of the discussion.

Page move?

I agree with those "partisans" - "quotient field" is a desperately confusing term! Would it not be preferable to move the page to one of the other two terms?

• "Quotient field" - 37,700 Google hits
• "Field of fractions" - 33,700 Google hits
• "Fraction field" - 27,100 Google hits
• "Field of quotients" - 10,800 Google hits (updated to add)

What do those who prefer the term "quotient field" use to refer to the "ring of fractions" where the underlying ring is not an integral domain? — ciphergoth 16:04, 23 February 2006 (UTC)

Well, since you ask... I suspect "quotient field" came about from talking about the "field of quotients", though the term seems to be even more rare acccording to google. Personally, I would prefer "field of fractions" or "field of quotients", but if it is not the most common term then there is little to be done about it other than put redirects from "field of fractions" and mention it in the text. The term is common and used, so whether or not it is confusing is somewhat besides the point. There is plenty of mathematical terminology that is confusing, both desperately and mildly so. I do not think there is a warrant to move from the (apparently, judging from your quick google survey) most common term to another term simply because we do not like the former or prefer the latter. While the text, speaking of "partisans" and the like, seems overly judgemental, other than redirects and perhaps a bit of cleaning up on this text I do not think a page move is warranted. Magidin 02:05, 26 February 2006 (UTC)

You make a good case, but let me have a go at bringing you around...

Where there is more than one term for something, Wikipedia tries to use one term consistently, and this is in itself exercising a little bit of wiggle room to teach an inconsistently-taught subject consistently. Wikipedia should certainly not make its own terms up or make idiosyncratic choices, but I think the difference in popularity is sufficiently small that we may allow other considerations to weigh in on what term Wikipedia should settle on, and which term is least confusing is a good consideration.

My Google survey shows that "quotient field" is used less than 35% of the time, and "field of fractions" less than 31% of the time; that's not a big difference. Also, "field of fractions" and "fraction field" are obvious variants of each other, so there's a case to be made that "quotient field" is the idiosyncratic choice.

Finally, we still need a term for "ring of fractions"... — ciphergoth 09:06, 26 February 2006 (UTC) (updated percentages to take "field of quotients into account — ciphergoth 11:57, 26 February 2006 (UTC))

One other point - how many of those Google hits for "quotient field" is Wikipedia responsible for? — ciphergoth 09:12, 26 February 2006 (UTC)

I think of this as the field of fractions, and 'quotient field' is unpleasantly ambiguous. Charles Matthews 10:01, 26 February 2006 (UTC)

I agree: when I hear "quotient field", I always check twice whether they're talking about the field of fractions, or about the quotient of a commutative ring by a maximal ideal, giving a field. I suspect that some of the Google hits for "quotient field" reported above refer to the second notion. AxelBoldt 17:08, 26 February 2006 (UTC)

OK, that's good enough for me - two of the three major participants like it, and the only argument against is a 4% difference in Google popularity, which may be misleadingly large for two reasons. In addition many Wikipedia pages already point to "field of fractions" and get redirected. I'll make the move. — ciphergoth 18:02, 26 February 2006 (UTC)

I know that this discussion took place three years ago and is long since settled, but if it is of any interest to anyone a count of a random sample from the first couple of hundred Google hits for "quotient field" suggests that about half of them mean the field of quotients or fractions, and the other half mean the quotient of a ring by an ideal. JamesBWatson (talk) 17:21, 5 March 2009 (UTC)

Integral domain?

I replaced "a zero-divisor free commutative ring with more than one element" with "an integral domain". JamesBWatson reverted the change, saying "No, if you read further down it explicitly states taht there need not be a 1. "Ring" is often taken as implying a 1, but not always." However, Wikipedia practice does currently define "ring" to have a multiplicative identity via monoid, so using this longer phrase doesn't make the more general applicability any clearer.

I think the best fix would be to start out discussing integral domains, and then discuss the more general case of a pseudo-ring later in the article, especially since AFAICT the field of fractions of a pseudo-ring which is not a ring doesn't necessarily contain the pseudo-ring, contrary to the assertion that it is "the smallest field in which it can be embedded". Thoughts? ciphergoth (talk) 13:15, 1 March 2010 (UTC)

In wiki, it is currently the convention that rings have identity. Also, as a matter of clarity, I think the first sentence should have "a zero-divisor free commutative ring with more than one element" replaced by "an integral domain". Then use the second sentence to say basically what it already says and explain that the field of fractions can be defined in a slightly more general context. Because the fact of the matter is most people coming to this article will be looking for the word "integral domain", and everybody else can wait until the second sentence to see the more general context. RobHar (talk) 14:59, 1 March 2010 (UTC)

I suggest we write "field of quotients of an integral domain with or without identity." The ring convention is I think flexible enough to handle a contrast, because it's meant to set the implicit definition. ᛭ LokiClock (talk) 06:58, 10 March 2012 (UTC)

I have rewritten the opening of the article in line with the above suggestions: I hope satisfactorily. Incidentally, I don't fully understand "the field of fractions of a pseudo-ring which is not a ring doesn't necessarily contain the pseudo-ring". Perhaps this refers to attempting to apply the field of fractions construction to a general "pseudo ring", i.e. any ring in the broad sense (without requiring a 1). I have never come across the concept of a "field of fractions" in this broad sense, and I am not aware of any way of constructing a field in this case. Certainly it is obvious that no construction can always produce a field containing an isomorphic copy of the base ring/pseudo-ring. Perhaps the remark was intended to refer to the ring of fractions, rather than field. If, however, we are referring not to this general situation but rather to the much more limited situation described in the article (commutative, no zero divisors, at least two elements) then the construction does produce a field, and there is a natural injection of the original "pseudo-ring" into this field: 0 is mapped to 0, and any other element x is mapped to x²/x. JamesBWatson (talk) 09:12, 3 March 2010 (UTC)

Very foolish of me to miss the natural injection - thanks! Made that intro slightly shorter by linking to the existing article on pseudo-rings (aka rngs). ciphergoth (talk) 14:03, 4 March 2010 (UTC)

I've added the general embedding for rngs from Hungerford's Algebra; the previous reference (Hartley and Hawkes) seemed only to contain the construction for (unital) rings, and did not give any embedding for rngs. --Frentos (talk) 05:43, 16 February 2018 (UTC)

Zero is sometimes a zero divisor

The element 0 of a ring R is a zero divisor if and only if R is not the zero ring. See Zero_divisor#Zero_as_a_zero_divisor for references and for an explanation of how the best definition leads to this. Ebony Jackson (talk) 07:18, 14 December 2013 (UTC)

"Citation needed" removed

I've removed the citation needed template from § See also. Citations don't go in that section; if you have questions about the Ore condition, please mark them on that page instead. Thanks, 128.135.98.218 (talk) 03:56, 23 October 2018 (UTC)

Universal Condition

The universal property doesn't seem right. I see no reason why the domain itself doesn't satisfy the requirements for Quot(R). Perhaps g should be a field monomorphism? --MarSch 28 June 2005 17:12 (UTC)

"Field monomorphism" is unnecessary; a ring homomorphism with domain a field is either the zero map or is one-to-one (and hence a monomorphism). Since h is injective, and g extends h, g cannot be the zero map. As for the "universal property", it's a field with the corresponding universal property (that is, the field, together with embedding of R into it, that has this property relative to embeddings of R into fields). That's why R itself doesn't satisfy it, unless it is already a field. 06:13, 24 October 2018 (UTC)