Endomorphism Rings .... Bland, Example 7, Section 1.1 .... ....

[Math Amateur]

I am reading Paul E. Bland's book, "Rings and Their Modules".

I am focused on Section 1.1 Rings and need some help to fully understand the proof of part of Example 7 on page 10 ... ...

Example 7 on page 10 reads as follows:View attachment 8197In the above example from Bland we read the following:

" ... ... \(\displaystyle \text{ End}_\mathbb{Z} (G) \) denotes the set of all group homomorphisms \(\displaystyle f \ : \ G \to G\) ... ... "Can someone explain exactly why \(\displaystyle \mathbb{Z}\) is in the symbol/notation \(\displaystyle \text{ End}_\mathbb{Z} (G) \) for the set of all group homomorphisms \(\displaystyle f \ : \ G \to G\) ... ... ?Peter

 

[steenis]

The $”R”$ in $Hom_R(M,N)$ denotes that we are working with $R$-modules and $R$-maps.
$Hom_{\mathbb{Z}}(G,H)$ denotes that we are dealing with $\mathbb{Z}$-modules and $\mathbb{Z}$-maps. Now $\mathbb{Z}$-modules are additive abelian groups and $\mathbb{Z}$-maps are homomorphisms between additive abelian groups.

$End_{\mathbb{Z}}(G)$ is short for $Hom_{\mathbb{Z}}(G,G)$ and the $”\mathbb{Z}”$ denotes that the underlying ring is $\mathbb{Z}$.

 

[Math Amateur]

The $”R”$ in $Hom_R(M,N)$ denotes that we are working with $R$-modules and $R$-maps.
$Hom_{\mathbb{Z}}(G,H)$ denotes that we are dealing with $\mathbb{Z}$-modules and $\mathbb{Z}$-maps. Now $\mathbb{Z}$-modules are additive abelian groups and $\mathbb{Z}$-maps are homomorphisms between additive abelian groups.

$End_{\mathbb{Z}}(G)$ is short for $Hom_{\mathbb{Z}}(G,G)$ and the $”\mathbb{Z}”$ denotes that the underlying ring is $\mathbb{Z}$.

Thanks for the help, steenis ...

Peter