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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:View attachment 6452

A ring homomorphism is defined by Lovett as follows:https://www.physicsforums.com/attachments/6453Thoughts so far ... ...

One ring homomorphism, \(\displaystyle f_1 \ : \ \mathbb{Z} \rightarrow \mathbb{Z}\) would be the Zero Homomorphism defined by \(\displaystyle f_1(r) = 0 \ \forall r \in \mathbb{Z}\) ...

(\(\displaystyle f_1\) is clearly a homomorphism ... )

Another ring homomorphism \(\displaystyle f_2 \ : \ \mathbb{Z} \rightarrow \mathbb{Z}\) would be the Identity Homomorphism defined by \(\displaystyle f_2(r) = r \ \forall r \in \mathbb{Z}\) ...

(\(\displaystyle f_2\) is clearly a homomorphism ... )

Now presumably ... ... ? ... ... \(\displaystyle f_1\) and \(\displaystyle f_2\) are the only ring homomorphisms from \(\displaystyle \mathbb{Z} \rightarrow \mathbb{Z}\) ... ... but how do we formally and rigorously show that there are no further homomorphisms ... ...

Hope that someone can help ...

Peter

=============================================================

Have no tried the following idea ... but no luck ...Suppose \(\displaystyle \exists \ f_3 \ : \ \mathbb{Z} \rightarrow \mathbb{Z}\) ... ... try to show no such \(\displaystyle f_3\) exists ... at least no \(\displaystyle f_3\) that is different from \(\displaystyle f_1 , f_2\) exists ... ...Let \(\displaystyle f_3(2) = x \) where \(\displaystyle x \in \mathbb{Z}\) ...Then \(\displaystyle f_3(4) = f_3( 2 \cdot 2 ) = f_3( 2) f_3( 2) = x^2 \)

and

\(\displaystyle f_3(4) = f_3( 2 + 2) = f_3( 2) + f_3( 2) = x + x = 2x\)

Then we must have \(\displaystyle x^2 = 2x\) ... ... ... (1)

I was hoping that there would be no integer solution to (1) ... but \(\displaystyle x = 0\) satisfies ... so ... problems ..

Maybe a similar approach with different numbers will work ... ...

Can anyone comment on this type of approach ...

Peter

I need some help with Exercise 1 of Section 5.4 ... ... ...

Exercise 1 reads as follows:View attachment 6452

**Relevant Definitions**A ring homomorphism is defined by Lovett as follows:https://www.physicsforums.com/attachments/6453Thoughts so far ... ...

One ring homomorphism, \(\displaystyle f_1 \ : \ \mathbb{Z} \rightarrow \mathbb{Z}\) would be the Zero Homomorphism defined by \(\displaystyle f_1(r) = 0 \ \forall r \in \mathbb{Z}\) ...

(\(\displaystyle f_1\) is clearly a homomorphism ... )

Another ring homomorphism \(\displaystyle f_2 \ : \ \mathbb{Z} \rightarrow \mathbb{Z}\) would be the Identity Homomorphism defined by \(\displaystyle f_2(r) = r \ \forall r \in \mathbb{Z}\) ...

(\(\displaystyle f_2\) is clearly a homomorphism ... )

Now presumably ... ... ? ... ... \(\displaystyle f_1\) and \(\displaystyle f_2\) are the only ring homomorphisms from \(\displaystyle \mathbb{Z} \rightarrow \mathbb{Z}\) ... ... but how do we formally and rigorously show that there are no further homomorphisms ... ...

Hope that someone can help ...

Peter

=============================================================

Have no tried the following idea ... but no luck ...Suppose \(\displaystyle \exists \ f_3 \ : \ \mathbb{Z} \rightarrow \mathbb{Z}\) ... ... try to show no such \(\displaystyle f_3\) exists ... at least no \(\displaystyle f_3\) that is different from \(\displaystyle f_1 , f_2\) exists ... ...Let \(\displaystyle f_3(2) = x \) where \(\displaystyle x \in \mathbb{Z}\) ...Then \(\displaystyle f_3(4) = f_3( 2 \cdot 2 ) = f_3( 2) f_3( 2) = x^2 \)

and

\(\displaystyle f_3(4) = f_3( 2 + 2) = f_3( 2) + f_3( 2) = x + x = 2x\)

Then we must have \(\displaystyle x^2 = 2x\) ... ... ... (1)

I was hoping that there would be no integer solution to (1) ... but \(\displaystyle x = 0\) satisfies ... so ... problems ..

Maybe a similar approach with different numbers will work ... ...

Can anyone comment on this type of approach ...

Peter

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