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Let $I_1, \dots , I_n$ be ideals of a ring $R$ with identity having zero intersection. Assume that for some $x\in R$, $x+I_ i$ is an element of the right socle of $R/I_ i$, for each $ i=1,\dots , n$. My question: "Is it necessarily true that $x$ belongs to the right socle of $R$?"

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There’s a natural injective module homomorphism $$R\to\bigoplus_iR/I_i$$ that takes $x$ into the semisimple submodule $\bigoplus_i\text{soc}(R/I_i)$, so the right ideal generated by $x$ is semisimple, and so $x\in\text{soc}(R)$.

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  • $\begingroup$ It seems that the same solution holds for arbitrarily many ideals $I_k$. $\endgroup$
    – karparvar
    Mar 1, 2018 at 15:29
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    $\begingroup$ @karparvar No, for infinitely many ideals the natural homomorphism is into the direct product, not the direct sum. If $R$ is the direct product of infinitely many copies of a field, and you take the kernels of projections onto the factors as your ideals, then that gives a counterexample. $\endgroup$
    – Jeremy Rickard
    Mar 1, 2018 at 15:53

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