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Let $\mathcal{A}$ be an abelian category. Call an object $M\in\mathcal{A}$ semisimple if every exact sequence $0\longrightarrow M'\longrightarrow M\longrightarrow M''\longrightarrow 0$ splits. Is it true, that the direct sum $\bigoplus M_i$, where all $M_i$ are semisimple, is semisimple again? I know the result is true for $R$-modules by the simple module characterization.

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Nope, this isn't true in general. For instance, let $\mathcal{A}=Ab^{op}$. Semisimplicity is a self-dual concept, so the semisimple objects in $\mathcal{A}$ are the same as the semisimple objects in $Ab$, which are the direct sums of groups of the form $\mathbb{Z}/p$ for $p$ prime. But, for instance, the group $\prod_{p\text{ prime}}\mathbb{Z}/p$, which is a direct sum of the semisimple objects $\mathbb{Z}/p$ in $\mathcal{A}$, is not semisimple (for instance, any element which is nonzero in infinitely many coordinates is not torsion, whereas any element of a semisimple abelian group is torsion).

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