I'm currently reading proof of Theorem $5.9$ from the paper The ideal structure of the Haagerup tensor product of $C^{\ast}$-algebras. Let $A$ and $B$ be $C^{\ast}$-algebras and $K$ be a closed ideal of $A\otimes^h B$ satisfying, for any pair of closed ideals $I$ and $J$ of $A\otimes^hB$ satisfying $I \cap J \subseteq K$ then either $I \subseteq K$ or $J\subseteq K$. The following is written in the proof.
Choose closed ideals $M$ and $N$ of $A$ and $B$ respectively which are maximal with respect to the property that $M \otimes^h B+ A\otimes^h N \subseteq K$.
I'm not able to follow this step. Can someone please explain how does one chooses closed ideals $M$ and $N$ satisfying the above relation.
A typical Zorm Lemma's argument works here.
Let $$ \mathcal F=\{(M,N):\ M\lhd A,\ N\lhd B,\ M \otimes^h B+ A\otimes^h N \subseteq K\}, $$ ordered by pointwise inclusion ($(M_1,N_1)\leq (M_2,N_2)$ means $M_1\subset M_2$, $N_1\subset N_2$). The family $\mathcal F$ is nonempty since $(\{0\},\{0\})\in\mathcal F$. Suppose that $\{(M_j,N_j)\}$ is a chain in $\mathcal F$. Then $(M_\infty,N_\infty)$, where $M_\infty=\bigcup_jM_j$ and $N_j=\bigcup_jN_j$, is an upper bound for the chain. Indeed, using that the tensor product is the closure of the algebraic product and that $K$ is closed, one gets $M_\infty \otimes^h B+ A\otimes^h N_\infty \subseteq K$. So $\mathcal F$ admits a maximal element $(M,N)$. And $M$ and $N$ are closed, for otherwise $(\overline M,\overline N)$ contradicts the maximality.