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 $(α_i)$ (\alpha_i)$ where $\alpha_i \in M_i$ and $α_i \alpha_i = 0$ 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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