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.

Post a comment or leave a trackback: Trackback URL.

Leave a Reply

Fill in your details below or click an icon to log in: Logo

You are commenting using your account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s

%d bloggers like this: