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I am reading Paul E. Bland's book, "Rings and Their Modules".
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 $$M = \bigoplus_{ \mathbb{N} } \mathbb{Z}$$ ... ...
Further I am having considerable trouble seeing how/why $$M \cong M \bigoplus M$$ ... ...
Now I am taking $$\bigoplus_{ \mathbb{N} } \mathbb{Z}$$ 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:
$$M = \bigoplus_{ \mathbb{N} } \mathbb{Z}$$
$$= \mathbb{Z} \bigoplus \mathbb{Z} \bigoplus \mathbb{Z} \bigoplus \ \ ... \ ... \ ... \ \bigoplus \mathbb{Z} \ \ ... \ ... \ ... ( \mathbb{N} \text{ copies } )$$
$$= \{ (z_\alpha ) \in \prod_{ \mathbb{N} } \mathbb{Z}_i \text{ where } i \in \mathbb{N} \}$$
$$= ?$$Can someone please confirm my analysis so far ... as far as it goes, anyway ...
Can someone also please explain and clarify the meaning of $$\bigoplus_{ \mathbb{N} } \mathbb{Z}$$, and further, demonstrate how Bland deduces that $$M \cong M \bigoplus M$$?
Needless to say, I do not follow the rest of the example ... ... so any help with that will also be appreciated ...
Peter
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 $$M = \bigoplus_{ \mathbb{N} } \mathbb{Z}$$ ... ...
Further I am having considerable trouble seeing how/why $$M \cong M \bigoplus M$$ ... ...
Now I am taking $$\bigoplus_{ \mathbb{N} } \mathbb{Z}$$ 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:
$$M = \bigoplus_{ \mathbb{N} } \mathbb{Z}$$
$$= \mathbb{Z} \bigoplus \mathbb{Z} \bigoplus \mathbb{Z} \bigoplus \ \ ... \ ... \ ... \ \bigoplus \mathbb{Z} \ \ ... \ ... \ ... ( \mathbb{N} \text{ copies } )$$
$$= \{ (z_\alpha ) \in \prod_{ \mathbb{N} } \mathbb{Z}_i \text{ where } i \in \mathbb{N} \}$$
$$= ?$$Can someone please confirm my analysis so far ... as far as it goes, anyway ...
Can someone also please explain and clarify the meaning of $$\bigoplus_{ \mathbb{N} } \mathbb{Z}$$, and further, demonstrate how Bland deduces that $$M \cong M \bigoplus M$$?
Needless to say, I do not follow the rest of the example ... ... so any help with that will also be appreciated ...
Peter