text:
Talk:Free module

Finite linear combination
I commented in this article (words to the effect) that for a module with an infinite free basis, every element is a convergent series of multiples of the basis elements.

This has recently been changed to state that it is a finite sum of multiples of the the basis elements.

Is this change correct?

-- Onebyone 23:56, 28 May 2004 (UTC)

 * Well, I guess we could define a topology on the module, but unless that is the case, we can't define convergence. Phys 05:34, 5 Dec 2004 (UTC)

 * The change is correct, even for topological modules. --129.70.14.127 00:03, 8 November 2007 (UTC)

 * Yes. There are unhappily two conflicting meanings of the word "basis" in use, as regards e.g. a Hilbert space, or other topological modules.  However, when deciding whether or not a module is free, the ordinary meaning (where we only consider finite linear combinations) is the one employed.

 * A related problem in the article is the fact that generating set now is linked to finitely-generated module, which of course is not very appropriate for an infinite basis. JoergenB (talk) 18:50, 22 August 2008 (UTC)

Definition
What is a "free basis"? Shouldn't it just be "basis"?--129.70.14.127 23:58, 7 November 2007 (UTC)
 * Done. --Beroal (talk) 15:35, 30 November 2010 (UTC)

Bad Introduction
I'm sorry to say that this article had a poor introduction. A free module is just a module that has a basis. Why start off talking about catagories and the like? It doesn't make sense. Nobody learns about vector spaces and modules from the catagory setting first. Everyone learns about the objects themselves, and once familiar we start to make our thinking more sophisticated by considering the bigger picture. This article was, for me, not at all usefull (even though I already knew what a free module was!) Dharma6662000 (talk) 21:06, 26 July 2008 (UTC)

 * Tere is one argument in favour of a category-theoretical introduction. In this way, the attribute "free" is put into its commonest(?) modern context from the beginning.  However, there are clearly arguments against, too.-JoergenB (talk) 18:40, 22 August 2008 (UTC)

 * I would like strongly to second the opinion voiced by Dharma6662000, extend the argument at which that post hints, and counter the one argument offered by JoergenB. With apologies, I will be alluding to certain other math articles as well; I have two excuses: first, many of the same people contributing to these other articles may be referring to this article; and second, I don't know of any specific one forum that is obviously more appropriate for venting a general complaint.

 * Let us call a topic “foundational” if it bridges the mathematics that is typically familiar to scientists and engineers (or undergrads) with the more modern topics and formulations (such as those that a typical math student, say in English-speaking countries, might not encounter until math graduate school). The property of “being a foundational topic” of course admits of a degree, and is somewhat a matter of taste, not to mention a matter contingent on idiosyncrasies in one's math education. Nevertheless, some topics clearly qualify as being foundational, e.g. tensor products, exterior algebras, tangent bundles, and, I think, even free modules. (Other topics presumably are clearly not foundational.)

 * Who is the most likely reader of an article on a “foundational” topic? It is not someone who is already initiated into category theory, at least not if they intend to learn something new. For in order to learn category theory in the first place, they had to have already learned about the objects covered in the article (as Dharma6662000 explained perfectly).

 * This brings me to JoergenB's argument, as follows: why would one want to put “things in modern context from the beginning,″ given the near certainty that this will render the article nearly useless to the uninitiated? The only plausible answer I can think of is this: the already-initiated might read an article on a foundational topic for a sort of aesthetic reason. They might like having Wikipedia (to borrow a phrase from Tolkien) “filled with things that they already knew, set out fair and square with no contradictions,″ and in accordance with the sensibilities prevailing among the cognoscenti.

 * I think the price for having such articles is a much-reduced usefulness to the people who need these articles the most, and I think this price is way too high. Please do not misunderstand me: the more sophisticated perspective should of course be covered in the article—but near the end, after the less abstract treatment. From the perspective of a likely reader of an article on a foundational topic, the insistence of a contributor on shoving the “abstract nonsense” in the reader's face practically from the first line is at best unhelpful, and at worst intimidating and irritating. Personally, it has already happened to me several times to have tried to look something up in Wikipedia, only to give up in frustration and look for guidance elsewhere—successfully, by the way, which confirms that the topic can be covered in a more pedestrian way than done in Wikipedia. Please, do not make a Wikipedia article understandable only to those who already know nearly everything that's in it.Reuqr (talk) 05:05, 27 February 2009 (UTC)

 * Introducing a free module via category theory seems ridiculous to me as well. Although, I can't say I am surprised to see it written this way -- category theory appears in excess within many Wikipedia math articles.  I think the subset of category theorists who are bent on describing every known mathematical idea via category theory are also the subset which most enjoy creating & modifying content on Wikipedia. Adammanifold (talk) 11:56, 19 May 2009 (UTC)

 * How can you define a free module as being a free module? In the first line... I'm not an expert at math, but that just seems really really bad! Perhaps if this isn't a mistake it could be clarified somehow. —Preceding unsigned comment added by 152.1.205.203 (talk) 20:43, 27 December 2009 (UTC)

 * Many facts about arrows could be taught at the elementary school level, instead of the nutty 19th century arithmetic curriculum that is taught today. This could be followed up by more challenging exercises at the high-school and college level.  The fact that there is a wide-spread revulsion to category theory is due to a failing of the educational system, and due to the fact that the current educational system insists on pounding arithmetic into kids heads, instead of math.  Thus, the concept of abelian groups, then groups, and vector spaces are all introduced as if they were generalizations of arithmetic.  If we taught arrows to children instead, most of the objections above would be moot.

 * In other words: to those objecting to category theory above: the problem is really that your education has failed you; this is not a fault of category theory, but rather the mindset of the entire academic system. Now that we actually *do* have a better way of explaining things, we should go for the new and better way, instead of sticking to the olde-fashioned approaches.

 * Err, well, uhm, maybe I have to eat my words; I see this article never contained more than a single sentence about free objects, so, yes, this is rather opaque. And worse, the article on free objects defines them as a generalization of free examples from many other fields (which is in fact how I learned them too), rather than as things in and of themselves, from first principles.   So we've got a generic problem here on WP.  I'm wondering if/how this could be fixed... linas (talk) 16:04, 2 September 2012 (UTC)

Some areas for improvement
The article does not mention:
 * projective modules, which under suitable restrictions are the locally free modules
 * free ideal rings, and the commutative specialization principal ideal rings
 * ideal class groups (measuring the difference between projective ideals and free ideals)
 * grothendieck groups (measuring the same for f.g. modules)
 * free resolutions, or the more popular finite free resolutions
 * freeness is not preserved under Morita equivalence, the property generated by freeness under Morita equivalence is again the notion of projective module
 * skew fields, the rings over which every module is free

The article's treatment of rank is poor. Rank should be given its own section.

For the zero ring, or rings without identity, free modules are very poorly behaved. For nonzero rings with identity: If E is a set, then every direct sum of E regular modules is determined up to isomorphism by the cardinality of E. If E, F are two such sets, E is infinite, and E and F have distinct cardinalities, then R^(E) and R^(F) are not isomorphic. Hence R^(E) has rank |E| if E is infinite. For some rings, R^m and R^n can be isomorphic modules without m being equal to n. This cannot happen for commutative rings, which should probably be mentioned. (Rings where R^m = R^n implies m=n, have the IBN property (linked in the article), and m is called the rank of R^m.

For the zero ring, the definition of free basis is slightly incorrect as it confuses a set (with its elements indexed) with an "indexed set", more commonly called a function from E to M. For nonzero rings, distinct e in E map to distinct elements of the module, but not for the zero ring (where all unital modules are free and isomorphic). JackSchmidt (talk) 19:40, 22 August 2008 (UTC)
 * Hungerford has a reasonable discussion of free modules over rings without identity, FWIW. They are basically free (unital) modules over that universal unital ring containing the non-unital ring. JackSchmidt (talk) 19:21, 8 May 2009 (UTC)

 * I have written a small section that gives pointers to the relation with projective and flat modules, but I think that more properties you mention (i.e. projectives being "almost" locally free) shouldn't be on this page but on the page for projective modules (or on another, corresponding, specialised page). Konrad (talk) 06:02, 11 May 2012 (UTC)

module
show that Q is not a finitely generated Z-module —Preceding unsigned comment added by 218.248.65.82 (talk) 06:35, 11 June 2009 (UTC)

"Every vector space is free"
This claim in the introduction is linked to a source which says "vector space" will mean a module over the complex numbers. So, even if the general result is true, I think the link should be removed, and the result should be proven in the article. — Preceding unsigned comment added by 98.248.228.230 (talk) 09:20, 27 November 2011 (UTC)

Cartesian product as a module
I know that this is a pretty minor point, but I disagree with these edits to regard the cartesian product of objects as implying a module structure: it is defined as a set, and is defined for any sets. When being careful, one should also not refer to the direct sum of copies of a noncommutative ring (not only does it collide with the terminology for the direct sum of rings, it is not clear whether a left- or right module is meant); one should refer to the "direct sum of n copies of a ring as left modules ". I believe that the left and right module structures that one might wish to attribute to a cartesian product are incompatible. I would argue that a module structure on the cartesian product of multiple copies of an arbitrary ring is unnatural (i.e. ill-defined), whereas the direct sum of n copies of a ring as a left module is well-defined. Authors like Bourbaki tend to be very careful with their wording in this regard. —Quondum 00:20, 20 May 2017 (UTC)

 * It is possible that I missed some subtleties but I didn't think the natural module structure there is tricky. I agree that we should be specific about whether R is viewed as a left or right module (basically the issue of Rn when R is not commutative) but after that, the module structure seeems clear; first, it is an abelian group by component-wise addition. The scalar multiplication is $$r \cdot (r_e | e \in E) = (rr_e | e \in E)$$ (if R is a left module). The key point, as I understand, a direct sum of free modules is free while a cartesian product of them might not be; that's why we use a direct sum. -- Taku (talk) 03:00, 20 May 2017 (UTC)

 * The root of this seems to be the use of "cartesian product" to mean the direct product. Since cartesian product has a clear definition in WP (that does not imply structure beyond what is implied by the set), I would prefer the use of the term "direct product" when this is meant.  Being someone who has first-hand experienced the amount of energy it takes to figure out when a term is used in a way that is not the defined meaning in WP (often requiring many-layered disambiguation), I guess I feel this more strongly than most.  I see that you have at least linked the correct meaning, but this has a quality of hidden meaning that seems to be unnecessary.  Some authors use terms loosely in this way (essentially forcing the reader to disambiguate through inference from context), but I think there is no reason to repeat such laziness here.  —Quondum 15:35, 20 May 2017 (UTC)
 * I do not understand well the object of this discussion. It seems that it is about the respective uses of "direct product", "Cartesian product" and "direct sum". As far as know, many algebraists consider "direct product" and "Cartesian product" as synonymous, for the following reason. Normally, "direct product" means product in category theory. It is an easy theorem that for all algebraic structures which are varieties of universal algebra (that is all axioms are identities), the direct product is the Cartesian product of the underlying sets, equipped with the component-wise operations. This is the case for groups, modules, rings, ..., but not for fields. This theorem should appear, but is lacking, in Direct product.
 * On the other hand, for modules, a direct sum is the same thing as a coproduct, but for rings, a coproduct is a tensor product. Thus, if $R$ is a ring, $$R^n$$ is, as a ring or left or right $R$-module, ..., the direct product of $n$ copies of $R$. But, as a ring, $$R^n$$ is not a direct sum of $n$ copies of $R$. Thus one should not talk of $$R^n$$ as a direct sum without clarifying that one considers only the module structure. In particular a free module has a natural ring structure only if and only its dimension is finite.
 * I hope that these remarks could be helpful for continuing to improve this article and other related articles. D.Lazard (talk) 18:00, 20 May 2017 (UTC)
 * This is a nuance of the way mathematicians use language (which, IMO, is shoddily, in the sense that adherent structure is often assumed while refusing to specify which structure is meant). What you have said here ("the direct product is the Cartesian product of the underlying sets, equipped with the component-wise operations") only implies that a direct product is a subcategory of a Cartesian product; this is like saying that a ring and a group are synonymous because a ring is a group equipped with a multiplication operation.  Consideration of the automorphism group of an object should make this clearer.  —Quondum 22:23, 20 May 2017 (UTC)

title: Talk:Free module


text: User talk:66.245.78.247

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: User talk:66.245.78.247


text:
North Lebanon

North Lebanon (شمال لبنان) is the northern region of Lebanon comprising the North Governorate and Akkar Governorate. On 16 July 2003, the two entities were divided from the same province by former Prime Minister Rafik Hariri. The division was known as Law 522.

The main cities and towns of the region are Halba, Tripoli, Miniyeh, Zgharta, Bcharreh, Amioun and Batroun. The districts of Akkar, Tripoli and Miniyeh-Danniyeh are known for their large Sunni Muslim population while the districts of Zgharta, Bcharreh, Koura and Batroun are known for their large Christian population.

Education
One of the best schools in Lebanon, International School - Al Koura, is in North Lebanon.

The University of Balamand (UOB) is in North Lebanon. UOB was ranked second-best university in Lebanon according to QS University Rankings 2023.

The most modern public Lebanese University campus is the North Campus.

Tourism
The charming historical town of Batroun is best known for its beach clubs, nightlife and seafood restaurants, while the medieval labyrinthine city of Tripoli is rich in history and architecture from the Mamluk era. At the northernmost tip of Lebanon lies the green Akkar region, with the beautiful lake area of Oyoun es-Samak and the attractive mountain slopes of Jabal Qammouah.

Qurnat as Sawda
Qurnat as Sawda is the highest point in Lebanon and the Levant.

Mseilha Walkway
The Mseilha Walkway (Darb Mseilha) is easily accessible from the Batroun highway, the entrance in next to the Mseilha Fort, a historic monument built by Fakhr al-Din II in the 17th century.

Horsh Ehden
Horsh Ehden Nature Reserve is located about three kilometers northwest of the center of Ehden. Visitors coming enjoy hiking, climbing and biking.

Tripoli Massacre
On 13 November 1943, a tragic incident took place in Tripoli, resulting in the loss of lives of 14 students, all under the age of 15. The students were struck by French tanks driven by Senegalese soldiers while participating in a peaceful march through the streets. During the demonstration, the students were fervently advocating for an end to the occupation. The massacre contributed to accelerating the declaration of Lebanon’s independence.

Background
One day before the massacre, the French occupation army arrested Abdul Hamid Karami and imprisoned him in the Rashaya Citadel.

Governorate Creation
On 12 June 1959, North Lebanon Governorate was created which separated Koura, Zgharta, Bcharreh and Batroun from Mount Lebanon.

Districts that formed North Lebanon:

1993-2003
On 23 November 1993, Minyeh and Danniyeh were separated from Tripoli and established their own district in North Lebanon.

2003-present
On 16 July 2003, Akkar was separated from North Lebanon due to the presence of government departments in the city of Tripoli which forced the citizens of Akkar to come from long distances in order to finish their government transactions.

Killings and massacres
On 12 February 1978, Syrian special units (الوحدات الخاصة السورية) attacked the northern village of Qnat, where they killed 15 Lebanese Forces fighters, the battle is known as battle of Qnat.

On 9 February 1986, Syrian Military Intelligence killed Tawhid leader Khalil Akkawi because he refused to fight the Lebanese Forces. Three supporters of Akkawi's Islamic Tawheed, or Islamic Unification Movement, were slain in gunfights with Syrian troopers after his burial, police reported.

Former Syrian minister of interior Mohammad al-Shaar participated in the 1986 Bab al-Tabbaneh massacre. In the 1980s, al-Shaar was a top intelligence official in northern Lebanon when Syrian troops stormed Tripoli and crushed a Sunni group that supported Palestine Liberation Organization chief Yasser Arafat.

Torture and detention
The Syrian military intelligence (المخابرات السورية) set up a torture prison inside the Evangelical School in Jabal Mohsen, the prison was called "American Prison" (معتقل الاميركان).

2021 Akkar explosion
On 15 August 2021, 28 people were killed and 79 injured when a fuel tank exploded in Akkar in North Lebanon.

2000
In the 2000 Lebanese general election, North Lebanon was divided in two electoral districts: Akkar-Danniyeh-Bcharreh and Tripoli-Miniyeh-Zgharta-Koura-Batroun. Those division were made by Ghazi Kanaan and Rafik Hariri to ensure that the Sunni majority would take away the true Christian opposition representation, however Nayla Moawad still managed to win.

2005
In the 2005 Lebanese general election, the March 14 alliance that was formed between Nayla Moawad, Boutros Harb, Kataeb, Lebanese Forces and Saad Hariri after the Cedar Revolution won all the seats in North Lebanon.

2009
In the 2009 Lebanese general election, North Lebanon was divided into 7 electoral districts: Akkar, Minniyeh-Danniyeh, Tripoli, Zgharta, Bcharreh, Koura and Batroun.

The March 14 Alliance won in 6 out of 7 electoral districts, the only electoral district they lost in was Zgharta which was won by the March 8 Alliance.

2018 and 2022
In the 2018 and 2022 Lebanese general election, North Lebanon was divided into 3 electoral districts: North I, North II and North III.

That electoral law was crafted to favor Gebran Bassil so he could finally win a seat after losing in 2005 and 2009.

Demographics
The following only includes numbers of northern Lebanese that are at least 21 years old and eligible to vote.

Notable events

 * Battle of Amioun
 * Bab al-Tabbaneh–Jabal Mohsen conflict
 * Chekka massacre
 * Ehden massacre
 * Battle of Qnat
 * 2007 Lebanon conflict
 * August 2013 Tripoli bombing
 * North Lebanon clashes (2014)
 * 2015 Tripoli, Lebanon bombings
 * 2021 Akkar explosion

title: North Lebanon


text:
User talk:Jim Horning

Hello and welcome to Wikipedia!

Here are some tips to help you get started:

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Good luck! Jrdioko

P.S. One last helpful hint. To sign your posts like I did above (on talk pages, for example) use the '~' symbol. To insert just your name, type (3 tildes), or, to insert your name and timestamp, use ~ (4 tildes).

Your Risks column in CACM
Hi, allow me to say a few words about your recent Risks column in CACM 12/2005. I'm not speaking for anyone besides myself here, and I don't really disagree with your specific findings. I just would like to point out that your points are not necessarily unique to Wikipedia, and that there already is an awareness of these issues within Wikipedia. In fact, some of your objections have been raised frequently enough to warrant a formulaic reply. Let me go over the main bullet points:

 * "Accuracy: You cannot be sure which information is accurate and which is not." This is true, and it's certainly a bigger problem for Wikipedia than for EB. This is a rather pervasive issue: the modern media in general (including TV, the Internet, newspapers, etc.) cannot be consulted on blind faith alone (people may wish to believe that that was once the case in the Good Old Days, before the invention of the telegraph, say, but they would of course be wrong). On the other hand, one cannot simply push the whole burden of evaluating sources back onto the reader. Which is why Wikipedia has a policy of verifiability and emphasizes the need to cite one's sources and check one's facts. Cynics could also point to the general disclaimer.
 * "Motives: You cannot know the motives of the contributors to an article." It should not be necessary to know or second guess the motives of contributors. There's a long list of things that Wikipedia isn't (it's not a blog, it's not a soapbox, it's not a forum for political campaigning, etc.). Contributions that ignore the goals of the project (which is, after all, to build an encyclopedia) can, and usually will, be reverted or removed. The editorial policy is to write articles in a neutral style that summarizes all relevant points of view (the so-called NPOV policy). What this entails is that point-of-view statements (even if they correspond to widely-held points of view) are out of place. So, for example, if someone with an agenda were to add a line like "ACME products rule/suck", it would get removed without much discussion. Acceptable (and much more informative) ways of saying the same thing include: "ACME products have received numerous awards, including...", or "Prof. John Q. Bumstead gave ACME products a failing grade in his review article ...".
 * "Uncertain Expertise"
 * "Some contributors exceed their expertise and supply speculations, rumors, hearsay," Information that is not verifiable cannot be included in Wikipedia. Anyone has the right to challenge any piece of information and demand to see sources.
 * "or incorrect information" That's the difficult part, at least in moderately obscure areas. Outright hoaxes are relatively easy to identify. It's the insidious vandals that can do some real damage by changing small pieces of information.
 * "It is difficult to determine how qualified an article's contributors are; the revision histories often identify them by pseudonyms, making it difficult to check credentials and sources." That's a general problem with online identities: reputation is established only in the context of the project, there is no other information to fall back on. On the other hand, genuine experts with credentials outside the project are welcome to identify themselves. There are quite a few notable people who have edited Wikipedia articles, including e.g. Joseph Kruskal (User:Kruskal), David S. Touretzky (User:Touretzky), Mitch Kapor (User:mkapor), Richard Stallman (User:rmstallman), etc.
 * "Volatility:"
 * "Contributions and corrections may be negated by future contributors." That is true and it happens all the time. Which is why Wikipedia is completely versioned, and all versions of an article can be referenced (see the "Permanent link" and "Cite this article" links in the toolbox).
 * "One of the co-authors of this column" – I'm guessing that would be you, as it wasn't Lauren Weinstein – "found it disconcerting that he had the power to independently alter the Wikipedia article about himself." First, thanks very much for correcting and expanding your own article. You doing this neither proves nor disproves your point. The near-universal ability to edit almost any article at any given time is morally neutral: it can be used for Good or for Evil. Wikipedia has a rather optimistic outlook, in that Jimmy Wales appears to believe that most people would rather use this power for Good (as you demonstrated). Also, if you were really up to no good, there are mechanisms in place to lock you out pretty quickly.
 * "Volatility creates a conundrum for citations: Should you cite the version of the article that you read (meaning that those who follow your link may miss corrections and other improvements), or the latest version (which may differ significantly from the article you saw)?" There's a pretty easy answer to that, as far as I can tell: all style guides, writer's manuals, etc. I'm aware of appear to agree that one should always cite the exact version of an article that one read. A reader who follows up on the citation can easily step through the version history of an article and even request a diff between the cited version and the most recent version. The situation is much better than the more common scenario, where a researcher reads the draft or pre-print of a journal article that's easily available online, whereas the journal article often isn't. In that situation, it will be much more difficult for another researcher to determine the differences between the informal pre-print or draft and the published version, whereas on Wikipedia the diff is just a few clicks away.
 * "Coverage:"
 * "Voluntary contributions largely represent the interests and knowledge of a self-selected set of contributors." Yup, there is systemic bias. At the very least there's an awareness that this is indeed a problem and a project to address it.
 * "They are not part of a careful plan to organize human knowledge." One of the premises of the Wikipedia project is that the traditional process of building an encyclopedia suffers from poor scalability. The current system, which allows virtually anyone to edit, has proven to be scalable (there are more than 600,000 registered accounts at this moment). Soliciting input from the interested public is nothing new; the Oxford English Dictionary, for example, was compiled, very slowly, on the basis of citations sent in by non-expert volunteers. While Wikipedia has not made any active efforts to recruit experts, there have been systematic efforts to identify gaps in the current coverage. For example, there is a long list of encyclopedia topics which lack articles.
 * "events that happened "before the Web" may be covered inadequately or inaccurately, if at all. More is written about current news than about historical knowledge." While this seems to be true as well, some would say it's more of a feature than a bug. The coverage of current events in EB is dismal, and Wikipedia has a clear advantage here. That's not to say that its coverage of the more distant past shouldn't be extended. Remember also that an encyclopedia is not necessarily intended to include all of human knowledge; all that's required is that it provide a good starting point for further research, by providing an overview of a topic, condensing the findings from the secondary literature, and providing pointers to primary and secondary references.
 * "Sources: Many articles do not cite independent sources." This may be true for a large portion of articles. However, Wikipedia has policies of verifiability and citing sources. To give just one example of how these policies are enforced, an article cannot be promoted to become a featured article unless it has a source and reference section.

None of this is to say that readers should have blind faith in what they read here. But the same is true for every medium and source: Think of the coverage of science and technology in the best newspapers of the world. People have made a hobby (and very useful in-class exercises) out of documenting the fallacies committed by science journalists. And I'm not talking about health coverage of local TV news, I'm talking about the New York Times, Washington Post, LA Times, etc. If you want to see an area where Wikipedia works very well as a non-specialist reference work, check out the mathematics articles. --MarkSweep (call me collect) 14:02, 2 December 2005 (UTC)

Response re: Risks column in CACM
Mark,

Most of these points were raised (many of them by me) in the give-and-take of the column's preparation. In our judgement, they did not raise Wikipedia to the level of trustworthiness usually associated with a reputable encyclopedia, nor provide the prospect of doing so in the foreseeable future. But you are free to place your trust where you wish.

I'd like to query just one of your points: "All style guides, writer's manuals, etc. I'm aware of appear to agree that one should always cite the exact version of an article that one read." This seems to me to be such an excellent principle that I wonder why Wikipedia uniformly violates it on internal references?

Jim H.

title: User talk:Jim Horning