 #1
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Homework Statement
I am reading Stephen Lovett's book, "Abstract Algebra: Structures and Applications" and am currently focused on Section 5.4 Ring Homomorphisms ...
I need some help with Exercise 1 of Section 5.4 ... ... ...
Exercise 1 reads as follows:
Homework Equations
The relevant definition in this case is as follows:
The Attempt at a Solution
Thoughts so far ... ...
One ring homomorphism ##f_1 \ : \ \mathbb{Z} \rightarrow \mathbb{Z}## would be the Zero Homomorphism defined by ##f_1(r) = 0 \ \forall r \in \mathbb{Z}## ...
(##f_1## is clearly a homomorphism ... )
Another ring homomorphism ##f_2 \ : \ \mathbb{Z} \rightarrow \mathbb{Z}## would be the Identity Homomorphism defined by ##f_2(r) = r \ \forall r \in \mathbb{Z}## ...
(##f_2## is clearly a homomorphism ... )
Now presumably ... ... ??? ... ... ##f_1## and ##f_2## are the only ring homomorphisms from ##\mathbb{Z} \rightarrow \mathbb{Z}## ... ... but how do we formally and rigorously show that there are no further homomorphisms ... ...
Hope that someone can help ...
Peter
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