- #1

Math Amateur

Gold Member

- 1,082

- 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:

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## ... ... ?

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 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 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