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Let $I = (f_1,f_2,\cdots, f_k) \subseteq \mathbb{C}[x_1,x_2,\cdots, x_m]$ and $J = (g_1,g_2,\cdots, g_r) \subseteq \mathbb{C}[y_1,y_2,\cdots, y_n]$ be radical ideals (we know that the $f_i$ and $g_j$ exist because $\mathbb{C}$ ring-adjoin any number of variables is Noetherian). Their concatenation is defined to be the ideal $$(f_1,f_2,\cdots, f_k, g_1,g_2,\cdots, g_r) \subseteq \mathbb{C}[x_1,x_2,\cdots, x_m, y_1,y_2,\cdots, y_n].$$ Is it possible that this concatenated ideal is not radical in $\mathbb{C}[x_1,x_2,\cdots, x_m, y_1,y_2,\cdots, y_n]$?

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    $\begingroup$ What if $m=n=1$? $\endgroup$
    – markvs
    Feb 7 at 3:06
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    $\begingroup$ See for example Tag 00I4. $\endgroup$ Feb 7 at 3:28
  • $\begingroup$ ah, so no counterexamples exist and the concatenated ideal is always radical? $\endgroup$
    – atenao
    Feb 7 at 6:49
  • $\begingroup$ @atenao Yes, that follows. $\endgroup$
    – Will Sawin
    Feb 10 at 16:03

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