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In contrast to the possibility of taking an arbitrary sequence of elements in the direct product of submodules , the definition for the direct sum of submodules of a module requires the indexed elements to vanish cofinitely(i.e. except finitely many times).

More precisely, Let $R$ be a ring, and ${M_i : i ∈ I}$ a family of left $R-$modules indexed by the set $I$. The direct sum of ${M_i}$ i.e. $\bigoplus M_i$ is then defined to be the set of all sequences $(\alpha_i)$ where $\alpha_i \in M_i$ and $\alpha_i = 0$ for cofinitely many indices $i$. (The direct product is analogous but the indices do not need to cofinitely vanish.)(Source: Wikipedia/Direct sum of modules.)We have similar definition for the sum of submodules.

I have not yet understood what pathology would be incurred or what simplification (if any) would be gained in assuming cofiniteness. Can you please explain this in simple terms (possibly, with examples) ?

Thanks.

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6 
 This should be asked on math.stackexchange.com IMO – Mariano Suárez-Alvarez Jan 25 2011 at 13:06
This is not a real question ... – Martin Brandenburg Jan 25 2011 at 13:17
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 en.wikipedia.org/wiki/Coproduct – Daniel Barter Jan 25 2011 at 13:31
@Daniel, thanks for your comment. But I have feeling that direct sums of modules predate coproducts(if I am right). So, that does not tell me the motivation for the necessity of cofiniteness in direct sums. I just do not see what motivated the definition. – To be cont'd Jan 25 2011 at 14:45
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 The motivation is that it satisfies the universal property for coproducts. This idea is independent of the words now used to formulate it. – Simon Wadsley Jan 25 2011 at 15:31
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closed as not a real question by Martin Brandenburg, Chandan Singh Dalawat, Angelo, Qiaochu Yuan, Andrea Ferretti Jan 25 2011 at 14:10

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