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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 Rmodules ... ...
Bland's Example on page 56 reads as follows:
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 2
In 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 Rmodules ... ...
Bland's Example on page 56 reads as follows:
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 2
In 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
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