I am trying to find three $R$-modules $V_{1},V_{2},V_{3}$ in $R=\mathbb{R}[x]/(x^{3}-x)$ as an $R$-algebra which have dimension 1 over $\mathbb{R}$ and $Hom_{R}(V_{i},V_{j})=\mathbb{R}$ if $i=j$, but $0$ if $i\neq j $ ?
What can we say about tensor product of $V_{i}$ and $V_{j}$ over $R$?
If we know that the $P:\mathbb{R}^{7} \rightarrow \mathbb{R}^{7}$ is projection onto a subspace of dimension 3 why does $P$ yields $R$-module structure on $\mathbb{R}^{7}$?
If we call the resulting module $M$ how can we describe the dimension over $\mathbb{R}$ of $Hom_{R}(V_{i},M)$ for all $i$.
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Multiplication by $x$ gives rise to a matrix like $A$ appearing in this old answer of mine. I'm sure better explanations exist on our site. Follow the links and search for more! – Jyrki Lahtonen Mar 29 '22 at 20:41
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Hint: Factor $x^3 - x$ and apply the Chinese Remainder Theorem. – Viktor Vaughn Mar 29 '22 at 23:32
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By CRT it means that we have list three $R^{3}$-modules which have the above properties but I couldn't find them – Divon Sardin Mar 30 '22 at 00:54
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@DivonSardin The CRT allows you to rewrite $\mathbb{R}[x]/(x^{3}-x)$ as the direct product of $3$ things. What are they? – Viktor Vaughn Mar 31 '22 at 06:09
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@ViktorVaughn Yes, We can write them as $\mathbb{R[x]}/(x) \times \mathbb{R}[x]/(x-1) \times \mathbb{R}[x]/(x+1) \cong \mathbb{R} \times \mathbb{R} \times \mathbb{R}$ – Divon Sardin Apr 01 '22 at 18:48
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Great, so these look like good candidates for $V_1$, $V_2$, $V_3$. – Viktor Vaughn Apr 01 '22 at 19:10
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@ViktorVaughn I think they might be $V_{1}=span<1,0,0>$, $V_{2}=span<0,1,0>$ and $V_{3}=span<0,0,1>$ but I'm not sure. – Divon Sardin Apr 02 '22 at 10:40