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At the behest of a comment on Mathoverflow, I am posting this here.

I am about to teach a few classes on modules and their operations, namely the following: direct product, direct sum, and finitely generated modules. The following is the "skeleton" version of how I plan to talk about them.

  1. The direct sum of modules $M_1, M_2$ is defined to be$$M_1 \oplus M_2 = \{(m_1, m_2) : m_1 \in M_1, m_2 \in M_2\}.$$The action of $A$ is given by $a(m_1, M_2) : = (am_1, am_2)$. More generally, for any set $I$ and any collection of $A$-modules $\{M_i\}_{i \in I}$, one has a direct product $A$-module$$\prod_{i \in I} M_i = \{(m_i \in M_i)_{i \in I}\}.$$We will often write an element of $\prod_{i \in I} M_i$ in the form $\sum_{i \in I} m_i$ instead of $(m_i \in M_i)_{i \in I}$.
  2. The $A$-module $\prod_{i \in I} M_i$ contains a submodule$$\bigoplus_{i \in I} M_i = \{(m_i \in M_i)_{i \in I} : m_i = 0 \text{ for all but finitely many }i\},$$called direct sum. In particular, for any $n \ge 1$, we write$$M^n := \underbrace{M \oplus \dots \oplus M}_{n \text{ times}}.$$
  3. The sum of a collection of submodules $M_i \subseteq M$, $i \in I$, is defined to be$$\sum_{i \in I} M_i = \{m_{i_1} + \dots + m_{i_j} : m_{i_j} \in M_{i_j}\}.$$
  4. The submodule generated by a fixed element $m \in M$ is defined to be$$Am = \{am : a \in A\}.$$More generally, for any collection $\{m_i\}_{i \in I} \subset M$, of elements of $M$, one has$$\sum_{i \in I} Am_i \subset M,$$the submodule generated by the $m_i$. We say that $M$ is finitely generated if there exist $m_1, \dots, m_k \in M$ such that $M = Am_1 + \dots + Am_k$.

What are some good things to add about these things which other people have found helpful when teaching/learning about these things?

Best, C

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    $\begingroup$ The title of this question sounds roughly answerable, but the text of the question seems to me to be very vague. Can you pin down more precisely what you are interested in knowing? $\endgroup$ – Jessica B Sep 23 '15 at 4:57
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    $\begingroup$ Do you have any applications of modules in mind, or are you just surveying algebraic structures? The former would probably dictate the sort of things you'd want to include (e.g. I only encountered modules in the context of group representations). $\endgroup$ – pjs36 Sep 25 '15 at 1:25
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    $\begingroup$ Concrete examples are always good to have, and applications where the viewpoint of some structure as a module actually makes something more easy to see. $\endgroup$ – user21820 Nov 2 '15 at 16:38
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    $\begingroup$ What is the interesting part meant to be? All I can see is a list of definitions that are familiar (hopefully) from other structures. What do you hope to gain from introducing modles? $\endgroup$ – Jessica B Dec 24 '15 at 21:36

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