Talk:Zero object (algebra)

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"isomorphic to the empty matrix"[edit]

I do not get to which exactly realization of the empty matrix the space of possible multiplications is isomorphic. My original version, defined by the empty matrix, already had an ambiguity because the multiplication is defined by a vector-valued matrix, or by 0×0×0 matrix. In any case there is only one empty matrix, does not matter the square one or the cubic. But assertion of the isomorphism without an exact definition of "empty matrix" is even less precise speech than my version. Incnis Mrsi (talk) 20:12, 27 February 2012 (UTC)

My real problem is to have an abstract algebra defined by a less abstract one, a matrix algebra. The object concerned (a trivial module) has already been adequately defined. I felt that the mention of matrices was illustrative only. An empty matrix may be given as an example, but perhaps that should not be in the lead. — Quondum 04:27, 28 February 2012 (UTC)

Is the trivial algebra a zero object?[edit]

Is the trivial algebra also a zero object in some category? I am not an expert in the category theory. Incnis Mrsi (talk) 20:12, 27 February 2012 (UTC)

Move[edit]

Current title "trivial module" raised an accusation in ambiguity. What backup names can we use if the problem become severe? I can propose zero space and trivial space, although the latter is likely an OR. Incnis Mrsi (talk) 20:12, 27 February 2012 (UTC)

I think that "null module" or "zero module" are more usual names for this notion. Moreover, it should be mentioned in the page that this module is frequently used in the context of exact sequences, where it is denoted simply by 0 (it appears twice in the short exact sequences). In any case one has to look on books of homological algebra to see how this module is named there. I believe "null module", but I may be influenced by the usual French denomination of "module nul". D.Lazard (talk) 23:05, 27 February 2012 (UTC)
Exact sequences' 0 may also denote the trivial Abelian group. But any Abelian group inherently is a ℤ-module, indeed. Incnis Mrsi (talk) 12:04, 28 February 2012 (UTC)
As with most mathematical terms, context seems to be significant. So whereas trivial group and probably trivial ring seem to be well-established, one evidently refers to a singleton set. So zero module may be an appropriate name for the module on a singleton set. The term trivial module does however occur, e.g. [1]. I do not understand the detail of objection particularly, perhaps we should check it first. Since we do seem to be attempting to address the article at the trivial objects of multiple types, perhaps we should do this properly, and rename it accordingly (and not simply by the most general case, which would be the singleton): sets, groups, modules, vector spaces, rings, algebras, and maybe others, as well as to give the dominant names for each. Interesting thought: fields apparently have no such object. — Quondum 05:16, 28 February 2012 (UTC)
I've located [2] with the interpretation of the objection. The ideal would be to get a practitioner in the field to comment on usage of terms. I am definitely not an expert. — Quondum 05:33, 28 February 2012 (UTC)
It should be noted that Mathworld has two articles on the same object, called either "trivial module" [3] or "zero module" [4]. Note also that the first page considers also a notion of "trivial module structure" which exists only for "non unitary modules". I think that we should apply WP:notability, which means here to consider only the trivial objects which are used in non trivial math, and to mention the other ones only by sentences like "some authors call this notion in this way". Thus I propose to
  • Rename the page Zero module
  • Modify the text accordingly
  • Mention that "trivial module" is also used for this notion
  • Mention (and even emphasize) that the zero module is the zero object of the category of the modules over a ring. In fact, historically, the theory of categories (especially abelian categories) was, for a large part, a generalization of the theory of modules which was needed for homological algebra. Thus the name "zero object" is derived from "zero module".
D.Lazard (talk) 11:49, 28 February 2012 (UTC)
A complete tangle. Proofwiki.org refer as "trivial module" to the thing with zero multiplication, analogous to our trivial representation and zero algebra. And Quondum suggests the term "zero module" for the zero object. IMHO we should choose the name not more recognizable, but less ambiguous. I am almost sure that the word "trivial" will ultimately be dropped in the title, and trivial module become a dab page. "Zero module" is a possible choice indeed, although I like "zero space" more. Incnis Mrsi (talk) 12:04, 28 February 2012 (UTC)
A tangle, yes. Zero space needs to be checked for standardness, but I guess it to be more used specifically as over a field and not a ring, and is thus may be a poor choice. We may be faced with many poor options (blame non-standardization of terminology). — Quondum 15:17, 28 February 2012 (UTC)

Zero object (algebra)[edit]

Yet another option could be moving to zero object (algebra), with mentioning the trivial group as the zero object of the category of groups. But it would make the article difficult to read for non-algebraists, and will shift the topic to category-theoretical stuff from current descriptive stance. Incnis Mrsi (talk) 16:36, 28 February 2012 (UTC)

I think this nevertheless has (plenty of) merit. It seems to get rid of the ambiguity in the title. We can make the content less technical if need be, introducing it as generally being the simplest object in any of several categories and that it is always a singleton, and then have a section for the corresponding object in each of several categories. It also gives scope for merging the small articles on each, such as trivial group and trivial ring. And then, at least, the naming problem in the case of a module is in the corresponding section, not in the name of an article. — Quondum 17:01, 28 February 2012 (UTC)
I put {{Merge}}s to that articles. May be this will increase an audience. Incnis Mrsi (talk) 18:28, 28 February 2012 (UTC)
Just a thought, and I'm sure this will come up in a more general discussion: parenthetical disambiguators are generally discouraged if there is nothing to disambiguate. In this instance, this means we'd have to consider zero object if we find no conflicting use for the term, for which there is currently a redirect that we would replace. — Quondum 19:27, 28 February 2012 (UTC)

I support the merge. However, care is needed for Zero object which redirects to Initial and terminal objects. Probably it should be moved to Zero object (category theory) and Zero object should become a dab page. But the relation is deeper and has to be made explicit in Zero object (algebra), as one may read in Initial and terminal objects: In the category of groups, any trivial group is a zero object. The same is true for the categories of abelian groups, modules over a ring, and vector spaces over a field. This is the origin of the term "zero object".D.Lazard (talk) 18:04, 29 February 2012 (UTC)

Does the term zero object have any use other than as used in category theory? The current redirect of zero object to Initial and terminal objects seem to me to be non-ideal (it mentions zero objects, but is not about them as such) and that in the cases mentioned, it is exactly what this article is supposed to be changed to refer to. The initial and terminal objects that don't conform (e.g. the empty set) are also not called zero objects. I'm weak in category theory, so the distinction between its use in algebra and in category theory escapes me. This will determine whether any disambiguator is appropriate. — Quondum 19:06, 29 February 2012 (UTC)
"Initial and terminal objects" may not be moved. There is a plenty of categories where objects similar to empty set (or, say, false (logic)) are initial objects but not terminal, and objects similar to singleton (mathematics) are terminal objects but not initial. Only in algebra these two typically coincide. Incnis Mrsi (talk) 19:23, 29 February 2012 (UTC)
I was not suggesting any change to that article; it is a very clear article and should not be touched. What I was suggesting is that the redirect zero object should be deleted to make way for this article, which should be renamed as zero object. — Quondum 19:30, 29 February 2012 (UTC)
But "zero object" is a well established terminology in category theory and something has to be kept for people who looks for information on it. This means a redirect zero object (category theory) and either a dab page zero object or hatnotes in both pages talking on "zero objects". D.Lazard (talk) 21:16, 29 February 2012 (UTC)
I have come across this definition here:
Definition 1.1.1 By a zero object in a category C, we mean an object 0 which is both an initial and a terminal object.
This defines it in terms of being initial and terminal, and so it is not a direct synonym for the trivial object that we are trying to document in this context (abstract algebra). (Forgive my obtuseness – I still need water wings.) I would then support the move of Zero object to Zero object (category theory), and the name of this article then is still open to debate. Option that occur to me are:
I've just found the article Zero element, and it looks like this article should be ciited as the main article for the section Zero element#Zero object. — Quondum 08:43, 1 March 2012 (UTC)
The problem is that "zero element" does not claim itself to be an article, but a dab page, so it has either be coverted to an article, or reformatted as a dab. Meanwhile, I proceed to the move and merger of "trivial ring", because it unlikely will be opposed. Some problems expected with trivial group because of different notation for the identity element. I will use the original name proposal (although do not insist that it is an unlimate solution), because "trivial object" is ambigous (see my reasonings about initial objects above), "Zero object (abstract algebra)" imposes a misleading contrast to abstract algebra (although our current content is quite traditional), and simply "zero object" is conflicting. Incnis Mrsi (talk) 09:25, 1 March 2012 (UTC)
D.Lazard made a quite inconvenient thing starting to alter the article independently, despite my forecast. Should I save my version over his one? Incnis Mrsi (talk) 11:34, 1 March 2012 (UTC)
Yes you should save your version over it: My edits are easy to restore after that, if they are yet needed (at least may edit of {{tl:Merge}}) will no more be useful). D.Lazard (talk) 11:49, 1 March 2012 (UTC)

{{mvar}} inside {{math}}[edit]

Take care with nesting {{mvar}} and {{math}}. It's evident that this is not intended use (only use one or the other), as one might observe from the following:

abcdefghijklmnopqrstuvwxyzABCDEFGHIJKLMNOPQRSTUVWXYZ
abcdefghijklmnopqrstuvwxyzABCDEFGHIJKLMNOPQRSTUVWXYZ
abcdefghijklmnopqrstuvwxyzABCDEFGHIJKLMNOPQRSTUVWXYZ

The last line (at least for me) is disproportionately large, whereas the first two lines are the same. — Quondum 19:19, 29 February 2012 (UTC)

Section Unital structures: rings[edit]

I'm a little confused. Unlike fields, the axioms defining a ring do not seem to exclude the trivial ring: it has a multiplicative identity element, and there is no requirement that 1≠0. It seems to be well-behaved with respect to the direct sum of rings, and I guess it should behave well as an initial object. So the argument given that ℤ is the initial object in the category of rings does not seem right. I know that Category of rings#Limits and colimits contradicts me. What have I got wrong? — Quondum 18:02, 2 March 2012 (UTC)

It is a pity that my wording was unable to convince even Quondum. So, I say the same in a less encyclopedic manner. The categorial "zeroness" of {0} actually relies on the fact that pseudo-rings, modules, and vector spaces are pointed sets with exactly one distinguished point 0; see my diagram. Inclusion of another constant 1 breaks all the picture, because object are immediately divided to "regular" (1 ≠ 0) and "pathological" (1 = 0), and any morphism becomes impossible from a pathological object to a regular one. By definition of the morphism. Incnis Mrsi (talk) 18:44, 2 March 2012 (UTC)
Your phrase "even Quondum" accords me more honour than I am worthy of. I can see the thrust of what you are saying in the wording used in the article. Thank you for this: I can see that I should focus more study on morphisms (and category theory generally). — Quondum 19:18, 2 March 2012 (UTC)

Unmerging "trivial ring"[edit]

Although the article Trivial ring (edit | talk | history | links | watch | logs) did not contain, as of February 28, almost anything which could justify a separate article, today I am not sure that redirecting it to here was a right solution. May be it is a discussion about unital structures what ought to constitute its topic, distinct from true zero objects. Should we restore the article and move the section to there? Incnis Mrsi (talk) 04:41, 3 March 2012 (UTC)

This seems to hinge on the implication that a trivial ring is not a true zero object (still beyond my ken). My support for the article's current name was based on the asssumption that these objects all fitted (and that zero object (algebra) was a distinct concept from zero object (category theory)). One option might be again to consider renaming this article more inclusively to encapsulate the "trivial object" concept. Alternatively, as you suggest, another article could be created to cover this topic (and zero objects would perhaps be classified under them, along with the trivial ring and related trivial objects). To have an article entirely devoted to the trivial ring while there are other similar "trivial objects" still does not seem right, though. — Quondum 07:02, 3 March 2012 (UTC)
You are correct in wondering whether the merge was the right solution, so I have undone it. The trivial ring article was easily understood by those whose algebra just about extends to groups, rings and fields but does not stretch to further abstraction. I have put it back for the time being --Rumping (talk) 22:27, 9 March 2012 (UTC)
It was neither polite nor practical to revert to a version of "trivial ring" criticized for absence of essential information, especially while this information is already present in a section of another article. One does not need to have good thoughts to just push the "undo" link in a web-interface. Good thoughts are required for improvement. Incnis Mrsi (talk) 07:59, 10 March 2012 (UTC)
I left the merge/discuss tag. But be clear that I think it is a error to do the merge. The zero object (algebra) article requires the reader to understand the concept of algebraic structure first and then consider the smallest example. The trivial ring article's key points are that {0} on its own, with ordinary addition and multiplication, is a ring and that if the additive and the multiplicative identity are the same then there is only one element, things far easier to understand. You may think that these points are repeated in the zero object (algebra) article, but they are not written so clearly and are obscured by more complicated ideas. --Rumping (talk) 22:53, 11 March 2012 (UTC)
So, what do you think is a right thing? An article about the trivial ring without mentioning its category-theoretical properties, eh? These trivial objects are not so easy as one could imagine. Since Rumping intervened to this complex matter (by undoing my changes), we should expect some constructive proposals, not only "merger is an evil". Incnis Mrsi (talk) 08:00, 12 March 2012 (UTC)
To Rumping: Before my recent edit, the trivial ring article was easily misunderstood by those whose algebra just about extends to groups, rings and fields, because the reader could believe that the trivial field is a field. I have done a similar edit in this page, but this fundamental information is not enough visible for the public intended by Rumping. I support the merge of trivial ring into zero object (algebra), but before merging the target should be rewritten to be accessible for the public addressed by Rumping. D.Lazard (talk) 09:00, 12 March 2012 (UTC)
My impression (which may be wrong) is that Incnis Mrsi does not think that the trivial ring is an example of a zero object (algebra), but that D.Lazard thinks that it is. I think that this should be resolved before the correctness of merging can be finalized. — Quondum 09:44, 12 March 2012 (UTC)
What do I think about it, was expressed in[5], as well as in the talk section above. If 1 belongs to the algebraic structure, then it is not an "example". If 1 is not an element of the structure, then it is. Incnis Mrsi (talk) 10:27, 12 March 2012 (UTC)
The fact that the trivial ring is not a zero object in the category of unital rings is not a real objection to call it also a zero ring. Thus it has a natural place in zero object (algebra): this page is not zero object (in a category). In other words, there is no mathematical disagreement between Incnis Mrsi and me. My point is that the considerations of theory of categories have to appear after the elementary considerations. Otherwise the article could be tagged as {{tl:Technical}}. D.Lazard (talk) 11:08, 12 March 2012 (UTC)
As someone who has tried to crystalize these concepts from WP I find this rather confusing. This article seems to draw somewhat on the concept of a zero object (in category theory), which is evidently not the same as the elementary concept of a zero object that this article seems to be addressing. The diff mentioned above seems to try to address this, but partly leaves the concept in the air with statements such as "depends on the precise definition". From the diff it would seem that the trivial ring is a zero object considered as a rng, but not when considered as a ring. This seems to match with the zero object (category theory), but that is not what this article is about. It also seems to suggest that the definition of rings requires that 1≠0. So I have to admit continued confusion, but I'm sure that between you a more easily understood presentation will emerge. — Quondum 13:14, 12 March 2012 (UTC)

"a" vs. "the" (for an object defined over a ring)[edit]

When speaking of "the trivial group" and "the trivial ring", the argument of isomorphism seems to be compelling. In the case of a module, a vector space or an algebra, the very definition of these categories implies the semantic of the associated scalar space ("over a ring"). This association seems to remain in the definition of the corresponding zero object, so that despite isomorphism in other respects, the operation of scalar multiplication is incompatible when the underlying scalars are not of the same type, and it makes less sense to regard all zero objects of these types to be the same object. I suggest that the article should refer to "a trivial module", "a trivial vector space" and "a trivial algebra" in recognition of this. Comment? — Quondum 09:14, 1 July 2012 (UTC)

I agree to refer to "a trivial module", "a trivial vector space" and "a trivial algebra" and "the trivial ring". On the opposite, for groups and similar structures, it depends on the context if the argument of isomorphism is compelling: Every group contains a trivial group, but the trivial subgroup of the integers may hardly be considered to be the same as the trivial subgroup of the rotations of a Euclidean vector space (for example). Isomorphism is not equality. On the other hand, for a stand alone trivial group, like in an exact sequence, there is no problem to use "the". D.Lazard (talk) 09:50, 1 July 2012 (UTC)
Even for groups, there can be different ones, which are only equal up to isomorphism. The trivial multiplicative subgroup is {1} and the trivial additive subgroup is {0} (say of the integers). These are not equal in the strictest sense because 1 is not equal to 0, and also because + and * do not extend the same way to the integers. But I think it would be fair to just mention this at the top and to state that "the" for the rest of the article means "the one unique up to isomorphism". But this is a universal issue in mathematics: when is equality "equality" and when is it just "isomorphism", and do we only use the definite article in the former case? The answer obviously depends on context, which is why you even hear people talk about "the singleton set", even when such a construction is obviously not unique. asmeurer (talk | contribs) 04:21, 16 December 2012 (UTC)