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 $R$?"


I appreciate any cooperation in answering my question!



From socle of quotients to socle of ring itself
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 $x\in\text{soc}(R)$.


