If $R$ and $S$ are rings with identity, is it true that socle is preserved under direct product, namely, is it true that $\mathrm{Soc}(R)\times \mathrm{Soc}(S)=\mathrm{Soc}(R\times S)$? (I mean by $\mathrm{Soc}(T)$ the direct sum of all the minimal right ideals of the ring $T$.)
I think that if the answer is positive, the assertion could be generalized to the $n$ case $R_1\times\cdots\times R_n$.
Thanks for any suggestion!
The minimal right ideals of $R_1\times\ldots\times R_n$ are precisely of the form $T_1\times\ldots\times T_n$ where $T_i$ is a minimal right ideal of $R_i$ for exactly one index, and the rest of the $T_j$ are zero.
If you sum all the minimal right ideals that are nonzero on index $i$, you get the right socle of $R_i$. If you sum all of the minimal right ideals, you get the right socle of the product ring. By this observation, your claim is true.