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Let $V$ be a complex vector space (i.e, over the filed of complex numbers) and $A$ be a complex algebra.

Suppose that $V$ is an $A$-module. Under what proper condition(s) there are irreducible submodules $V_i\leq V$ satisfying the following conditions?

  1. $V_i\cap V_j=0$,
  2. $V$ is embedded into $\prod V_i$
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    $\begingroup$ What do you mean by 2? Does the embedding have the restrict to the natural inclusions $V_i \hookrightarrow \prod V_j$? It seems more natural to ask for the natural map $\bigoplus V_i \twoheadrightarrow V$ to be surjective (or use quotients instead of subobjects and dualise). $\endgroup$
    – R. van Dobben de Bruyn
    Commented Jan 7, 2021 at 16:53
  • $\begingroup$ Yes, by the natural inclusion $V_i \hookrightarrow \prod V_j$ for every $i$. $\endgroup$
    – ABB
    Commented Jan 7, 2021 at 17:04
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    $\begingroup$ You did not answer my question. Are you requiring that the restriction of said embedding agrees with the natural inclusion? $\endgroup$
    – R. van Dobben de Bruyn
    Commented Jan 7, 2021 at 17:06

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