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We really should include the universal property of direct sums here, instead of having to dig it up in the category theory section. We never actually define what it *is* in this article, just give examples of what it can be in certain categories. 129.2.164.54 (talk) —Preceding undated comment added 20:26, 15 July 2010 (UTC).
After reading this article I still wonder what a direct sum is. I would like to see an example where the direct sum is calculated for some simple set and where all the steps are included. I'm so tired of mathematicians writing for mathematicians. —Preceding unsigned comment added by 62.20.172.163 (talk) 08:38, 20 March 2011 (UTC)
I also find this article confusing, and am particularly annoyed by initial references to Category theory. While there is no harm in mentioning category theory farther down in the article, initial references to it almost always obfiscate the obvious. — Preceding unsigned comment added by 146.186.131.40 (talk) 20:39, 29 September 2011 (UTC)
Here are my reactions:
1. It might be helpful for the article to make clear at the top that the term comes from abstract algebra.
2. Minimum necessary dependencies on other terms from abstract algebra should be made, and where unavoidable, they should be referenced properly.
3. If at all possible, postpone reference to category theory until the body of the article, because readers seeking to understand the term ‘direct sum’ will not all have heard of category theory.
4. The article compares poorly to the clearer, more informative article found here: http://mathworld.wolfram.com/DirectSum.html
5. The difference between direct sum and direct product needs to be explained step-by-step in the simplest possible terms, including the convention of using the + symbol as far as possible only for a commutative operation
Jonathan G. G. Lewis 07:54, 23 October 2013 (UTC) — Preceding unsigned comment added by Jonazo (talk • contribs)
Imagine I am an undergraduate student coming here to understand the definition of a direct sum of matrices from my linear algebra course (arguably a much more common use case than afficionados of category theory). Reading this article, I might conclude that `direct sum' is a term reserved for much more abstract algebraic objects. Would it hurt to have a simple mention that direct sum also refers to matrices (i.e., block diagonality???). Simply baffling... — Preceding unsigned comment added by 134.60.158.127 (talk) 14:03, 20 February 2014 (UTC)
I have some trouble with this line: " the direct sum R ⊕ R {\displaystyle \mathbf {R} \oplus \mathbf {R} } \mathbf{R} \oplus \mathbf{R} , where R {\displaystyle \mathbf {R} } \mathbf{R} is real coordinate space, is the Cartesian plane, R 2 {\displaystyle \mathbf {R} ^{2)) {\displaystyle \mathbf {R} ^{2))". If R is understood as span(v), with v a one dimensional vector, then shouldn't span(v) ⊕ span(v) equal span(v)? — Preceding unsigned comment added by Vaglame (talk • contribs) 18:12, 4 November 2018 (UTC)
I fixed the lead, but the rest of the article needs work. Ebony Jackson (talk) 05:36, 1 January 2014 (UTC)
Many other direct sums (groups, rings, and vector spaces to mention just three) are at least as important as direct sums of modules. I still remember learning ring theory out of Lambek Lectures on Rings and Modules and while it was mathematically sound, it confused the hell out of me as an undergraduate. Rick Norwood (talk) 13:46, 27 January 2014 (UTC)
The difference between direct sum and direct product is analogous to the difference between product topology and box topology: both direct sums and product topologies have a cofiniteness condition that direct products and box topologies don't have. GeoffreyT2000 (talk) 00:42, 17 February 2015 (UTC)
Is it time to remove the 'rewrite needed' tag? It was added in August 2012, since when the article pretty much has been rewritten. Mortee (talk) 03:49, 22 January 2017 (UTC)
I don't think that Z6 example is correct, since {1, 3, 5} isn't a subgroup, and the product listed contains redundant elements. But I'm not an expert. — Preceding unsigned comment added by 173.127.238.133 (talk) 07:03, 21 October 2018 (UTC)