Consider two analytic algebras $\mathcal{O}_{\mathbb{C}^{m},0}/\mathcal{I}_{A}$ and $\mathcal{O}_{\mathbb{C}^{n},0}/\mathcal{I}_{B}$, where $\mathcal{I}_{A}:=\mathcal{O}_{\mathbb{C}^{m},0}f^{1}_{0}+\ldots+\mathcal{O}_{\mathbb{C}^{m},0}f^{k}_{0}$, $\mathcal{I}_{B}:=\mathcal{O}_{\mathbb{C}^{n},0}g^{1}_{0}+\ldots+\mathcal{O}_{\mathbb{C}^{n},0}g^{l}_{0}$ for $f^{1},\ldots,f^{k}\in\mathcal{O}_{\mathbb{C}^{m}}(D)$, $g^{1},\ldots,g^{l}\in\mathcal{O}_{\mathbb{C}^{n}}(E)$. Define the ideal $\mathcal{I}_{AB}$ of $\mathcal{O}_{\mathbb{C}^{m+n},(0,0)}$ given by $$\mathcal{I}_{AB}:=\mathcal{O}_{\mathbb{C}^{m+n},(0,0)}(f^{1}\circ\pi_{m})_{(0,0)}+\ldots+\mathcal{O}_{\mathbb{C}^{m+n},(0,0)}(f^{k}\circ\pi_{m})_{(0,0)}+\mathcal{O}_{\mathbb{C}^{m+n},(0,0)}(g^{1}\circ\pi_{n})_{(0,0)}+\ldots+\mathcal{O}_{\mathbb{C}^{m+n},(0,0)}(g^{l}\circ\pi_{n})_{(0,0)},$$ where $\pi_{m},\pi_{n}$ are canonical projections from $\mathbb{C}^{m+n}$ to $\mathbb{C}^{m}$ and $\mathbb{C}^{n}$ respectively.
If $\mathcal{O}_{\mathbb{C}^{m},0}/\mathcal{I}_{A}\cong \mathcal{O}_{\mathbb{C}^{m'},0}/\mathcal{I}_{A'}$ and $\mathcal{O}_{\mathbb{C}^{n},0}/\mathcal{I}_{B}\cong \mathcal{O}_{\mathbb{C}^{n'},0}/\mathcal{I}_{B'}$ as analytic algebras, does this mean $\mathcal{O}_{\mathbb{C}^{m+n},(0,0)}/\mathcal{I}_{AB}\cong \mathcal{O}_{\mathbb{C}^{m'+n'},(0,0)}/\mathcal{I}_{A'B'}?$ Here $\mathcal{I}_{A'},\mathcal{I}_{B'}$ and $\mathcal{I}_{A'B'}$ are same as $\mathcal{I}_{A},\mathcal{I}_{B}$ and $\mathcal{I}_{AB}$ respectively.