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I am reading Paul E. Bland's book "Rings and Their Modules ...
Currently I am focused on Section 2.2 Free Modules ... ...
I need some help in order to fully understand Bland's Example on page 56 concerning directly finite and directly infinite R-modules ... ...
Bland's Example on page 56 reads as follows:View attachment 5615
Question 1
In the above Example from Bland's text we read the following:
" ... ... If $$M = \bigoplus_\mathbb{N} \mathbb{Z}$$, then it follows that $$M \cong M \oplus M$$ ... ... "How ... exactly ... do we know that it follows that $$M \cong M \oplus M$$ ... ... ?
Question 2In the above Example from Bland's text we read the following:" ... ...$$R = \text{ Hom}_\mathbb{Z} (M, M) \cong \text{ Hom}_\mathbb{Z} (M, M \oplus M )$$$$\cong \text{ Hom}_\mathbb{Z} (M, M) \oplus \text{ Hom}_\mathbb{Z} (M, M)$$ $$\cong R \oplus R$$ ... ... "
Although the above relationships look intuitively reasonable ... how do we know ... formally and rigorously that:$$\text{ Hom}_\mathbb{Z} (M, M) \cong \text{ Hom}_\mathbb{Z} (M, M \oplus M )$$$$\cong \text{ Hom}_\mathbb{Z} (M, M) \oplus \text{ Hom}_\mathbb{Z} (M, M) $$
Hope someone can help ...
Peter
Currently I am focused on Section 2.2 Free Modules ... ...
I need some help in order to fully understand Bland's Example on page 56 concerning directly finite and directly infinite R-modules ... ...
Bland's Example on page 56 reads as follows:View attachment 5615
Question 1
In the above Example from Bland's text we read the following:
" ... ... If $$M = \bigoplus_\mathbb{N} \mathbb{Z}$$, then it follows that $$M \cong M \oplus M$$ ... ... "How ... exactly ... do we know that it follows that $$M \cong M \oplus M$$ ... ... ?
Question 2In the above Example from Bland's text we read the following:" ... ...$$R = \text{ Hom}_\mathbb{Z} (M, M) \cong \text{ Hom}_\mathbb{Z} (M, M \oplus M )$$$$\cong \text{ Hom}_\mathbb{Z} (M, M) \oplus \text{ Hom}_\mathbb{Z} (M, M)$$ $$\cong R \oplus R$$ ... ... "
Although the above relationships look intuitively reasonable ... how do we know ... formally and rigorously that:$$\text{ Hom}_\mathbb{Z} (M, M) \cong \text{ Hom}_\mathbb{Z} (M, M \oplus M )$$$$\cong \text{ Hom}_\mathbb{Z} (M, M) \oplus \text{ Hom}_\mathbb{Z} (M, M) $$
Hope someone can help ...
Peter