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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!

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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.

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    Is it true for arbitrary direct products? Or, at least, is it true that the direct product of "socles" of rings falls into the socle of direct product of the rings? If $T$ is a minimal right ideal of the direct product of the rings, then some element $t\in T$ is nonzero, so some component of $t$, say, the $i$'th is nonzero. If we consider the tuple $(r_i)$ which comprises of zeros but at $r_i=1$, then $T(r_i)$ must be equal to $T$. So, we deduce that all the minimal right ideals of the product are of the form $\prod T_i$ as in your answer..... . – 2017-01-15
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    @karparvar No, what you suggest is trivially false. $\prod_{i=1}^\infty F$ is a counterexample. – 2017-01-16
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    So, the direct product of the socles is $\prod F$ itself, but, how do you calculate the socle of $\prod F$? I think it becomes $\oplus F$, but why? – 2017-01-17
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    @karparvar If I abbreviate the minimal ideals which are just the nonzero elements on a fixed coordinate as $F_i$, then yes, the socle contains all of the $F_i$. And then also, any other minimal ideal must be nonzero on some coordinate, say $j$, and then $S=SF_j=F_j$. So yes, this is the socle. Not sure if it applies in general, though. – 2017-01-17
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    Do you mean "socle" by $S$? – 2017-01-17
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    @karparvar No, I mean $S$ is a minimal ideal ( I was thinking of "simple module") I am obviously not proving the socle is simple. – 2017-01-17