A question in verifying conditions of Zorn' s Lemma

Question

I am studying algebra from Thomas Hunger Ford and I have a question in a thoerem on page $194-195$ of Chapter Modules.

I am adding it's image.

Edit :Question : How can I prove that every chain has a maximal element? ( For this I need to find 1 element and prove it maximal).

Any clue would be really appreciated.

Answer 1

Given a chain $\{h_i\}_{i\in I}$ define a function $h$ like this:

$1.$ Define Dom$(h)=\cup_{i\in I}$ Dom$(h_i)$.

$2.$ For every $x\in $Dom$(h)$ there is some $i\in I$ such that $x\in $Dom$(h_i)$. Then define $h(x)=h_i(x)$.

Now there are a few things that you need to check: that $h$ is well defined (what if there are two distinct $i,j\in I$ such that $x\in $Dom$(h_i)\cap $Dom$(h_j)$? You need to show that in that case $h_i(x)=h_j(x)$), that it is an element of $S$ and that it is an upper bound of the chain. All of the above should be easy to check, so I'll leave it to you. Of course you have to use the fact that $\{h_i\}_{i\in I}$ is a chain and not just some random subset of $S$.