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 $$\mathbb{C}[x_1,x_2,\cdots, x_m, y_1,y_2,\cdots, y_n]$$?
• What if $m=n=1$? Feb 7 at 3:06