Talk:Direct sum of modules

message: Although the text refers to the direct sum of abelian groups, it does not discuss the direct sum of non-abelian groups (let alone semi-direct sums or subdirect products, etc. of same).
Although as time has gone by, I've learned enough to enjoy the "big picture" of algebra, I needed something a bit more specific when I was going around the block the first time. And category theory is still a bit far off in the mists for me.
Perhaps in this case, a "direct sum of groups" page should be added which can back link to this page, which could be retitled something like "Algebraic direct sums" or just stay "direct sums". I could make a similar comment about direct product which is currently (mostly) a "direct product of groups" page. Any thoughts?  chas 19:39 7 oct 02 (UTC)
:As I mentioned in one of the edits, "direct sum of groups" really should redirect to "free product of groups". Revolver 18:32, 30 August 2005 (UTC)
thread_title:  Other direct sums 

message: I've only taken a linear algebra course so I found the article rather incomprehensible. My definition of a direct sum goes like this: Let V be a v.sp. and U1,...,Um be subspaces of V. Then V is a direct sum denoted V = U1 (+) ... (+) Um iff each v ∈ V can be written uniquely as a sum u1 + ... + um st each ui ∈ Ui.
Comments welcomed! Goodralph 23:20, 11 May 2004 (UTC)
thread_title: Linear algebra

message: I'm a bit confused as to the relationship between the direct sum and the Cartesian product. It appears that if I have two abelian groups, (G, *_G) and (H, *_H), I can take the direct sum of those groups to be 
:(G, *_G) \oplus (H, *_H) = (G\times H, *_{G\oplus H})
where *_{G\oplus H} is defined by
:u *_{G\oplus H} v := (u_G *_G v_G) \oplus (u_H *_H v_H).
Is that correct? If so, should "direct sum" be mentioned on the Cartesian product page? —Ben FrantzDale 14:24, 2 January 2007 (UTC)
replies: text: seems that the issue is just semantics. as your notation indicates, the underlying set is the Cartesian product of sets, on which you define an operation, and call this pair the direct sum of the two modules (or the direct product of groups, etc). On the other hand, one also speaking of the Cartesian product of two algebras, where the algebra operations are defined in a similar way on the Cartesian product of sets. In that context, the direct sum refers to just the resulting module structure. Mct mht 23:23, 4 January 2007 (UTC)
thread_title:  Direct sum  & Cartesian product 

message: I changed  "The cartesian product V × W can be given the structure of a vector space K" to "The cartesian product V × W can be given the structure of a K-vector space", thinking it uncontroversial, as I have never seen the former wording used before, and found it actually confusing (the K is in apposition to the "vector space").  That was reverted so I've changed again to "vector space over K", which is pretty standard I think.  Richard Pinch (talk) 06:26, 11 October 2008 (UTC)
thread_title:  Vector space terminology 

message: The procedure described for constructing the "direct sum" of algebras actually gives the product in the category of algebras.
Thus it should really be called direct product of algebras instead.
When one hear the "direct sum", one expects the coproduct in the category, which is the free product in the case of algebras, and not the direct product.
The terminology "direct sum of algebras" should therefore be avoided, although it is very widespread, especially in Lie theory. I suggest these things should be pointed out, including perhaps a proof (for people in doubt) that the direct product of two algebras does not satisfy the requirement of a coproduct in the category of algebras.
Fistel (talk) 18:42, 22 April 2010 (UTC)
replies: text: Thank you Fistel for the remarks on category language. Your frank statement that the terminology is "very widespread, especially in Lie theory" admits that this encyclopedia reflects common usage. Your note concerning "one expects the coproduct in the category" shows your concern for a particular kind of reader, not the general reader. Your note is important as mathematics communication evolves on the basis of attending to such matters. For the moment I am putting Joseph Wedderburn as a reference to confirm the present usage.Rgdboer (talk) 20:42, 22 April 2010 (UTC)
thread_title:  "Direct sum" of algebras 

message: I was reading an article by Nashed&Wahba(1974) "Generalized inverses in reproducing kernel spaces..." in SIAM J.Math.Anal., #5 — where the authors consider a linear operator A between Hilbert spaces X and Y. They immediately make the following claim: "It is well known and can be easily shown that if A is a bounded operator, or if A is a densely defined closed operator, then [generalized inverse] A† exists on , where ℜ(A) is the range of A." So I don't seem to understand the difference between this direct sum , and the entire space Y — could someone clarify this?  // stpasha »  04:39, 25 June 2010 (UTC)
thread_title:  Direct sum of a subspace and its orthogonal complement?