Here's a question I've been thinking about, it's a curiosity that I don't know how to answer. There could be a simple counterexample, or it could be true and I don't know how difficult it would be to prove.
If we fix $m$, is it always possible to find a sufficiently large $n$ satisfying the conditions of the following question: [Note: My original question was to determine whether it is true for arbitrary $m,n$ which was answered below negatively; I have edited it to make the question more interesting].
Define $ phi: S_m rightarrow S_{m+n}$ is a canonical embedding, and $phi^{*} : F[S_m] rightarrow F[S_{m+n}]$ and similarly embeddings $theta: S_{n} rightarrow S_{m+n}$, and the induced map, such that $phi(S_{m}) times theta(S_{n})$ is a direct product.
Given an element $x in phi^{*}(F[S_m]), x neq 0$, is it necessary that there exist an element $x' in F[S_{m+n}]$ so that the product $xx' in theta^{*}(F[S_n]), xx' neq 0$. It seemed true in the cases that I have tried, but they are quite small so I'm not certain if this is true.
Making the assumption $ text{char} F = 0$ would make it easier I'm sure, but even in this case I can't prove it.
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