- #1
Math Amateur
Gold Member
MHB
- 3,992
- 48
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:
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:
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