## The union of a chain of monomial ideals is a monomial ideal

Let $F$ be a field, let $R = F[x_1, \ldots, x_m]$, and let $K$ be a linearly ordered set. Let $\mathcal{M} = \{M_k\}_{k \in K}$ be a family of monomial ideals in $R$ such that if $i \leq j$ then $M_i \subseteq M_j$. Prove that $\bigcup_K M_k$ is a monomial ideal in $R$.

We showed in this previous exercise that $\bigcup_K M_k$ is an ideal of $R$. Thus it suffices to show that $\bigcup_K M_k$ has a monomial generating set.

To that end, let $G_k$ be a monomial generating set of $M_k$. Then $\bigcup_K M_k = \bigcup_K (G_k)$ $= (\bigcup_K G_k)$, so that $\bigcup_K G_k$ is a monomial generating set for $\bigcup_K M_k$.