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I am trying to understand Section 2.2 on free modules and need help with Example 5 showing a module with two bases ... ...

Example 5 reads as follows:View attachment 3569I am having trouble understanding the notation and meaning of \(\displaystyle M = \bigoplus_{ \mathbb{N} } \mathbb{Z}\) ... ...

Further I am having considerable trouble seeing how/why \(\displaystyle M \cong M \bigoplus M\) ... ...

Now I am taking \(\displaystyle \bigoplus_{ \mathbb{N} } \mathbb{Z}\) to be an external direct sum ... ...

Bland defines an external direct sum as follows:https://www.physicsforums.com/attachments/3570So ... ... following the above definition (at least I think I am correctly following it ...) we have:

\(\displaystyle M = \bigoplus_{ \mathbb{N} } \mathbb{Z}\)

\(\displaystyle = \mathbb{Z} \bigoplus \mathbb{Z} \bigoplus \mathbb{Z} \bigoplus \ \ ... \ ... \ ... \ \bigoplus \mathbb{Z} \ \ ... \ ... \ ... ( \mathbb{N} \text{ copies } )\)

\(\displaystyle = \{ (z_\alpha ) \in \prod_{ \mathbb{N} } \mathbb{Z}_i \text{ where } i \in \mathbb{N} \}\)

\(\displaystyle = ?\)Can someone please confirm my analysis so far ... as far as it goes, anyway ...

Can someone also please explain and clarify the meaning of \(\displaystyle \bigoplus_{ \mathbb{N} } \mathbb{Z}\), and further, demonstrate how Bland deduces that \(\displaystyle M \cong M \bigoplus M\)?

Needless to say, I do not follow the rest of the example ... ... so any help with that will also be appreciated ...

Peter