Talk:Direct sum of modules

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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)

[edit] Linear algebra

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)

That's an internal direct sum; you are already provided with the ambient V. Charles Matthews 05:45, 12 May 2004 (UTC)

[edit] Direct sum & Cartesian product

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)

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)
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