Question regarding a ring homomorphism.

Question

I have the following questions. I define a homomorphism $\phi:R\to R\times \mathbb{Z}$ by $\phi(r)=(r,0)$.
I can't understand the following:

1.) My notes describe that this homomorphism is not onto. but I can't understand why.

2.) It states $R\cong \phi(R) \subseteq R\times \mathbb Z $. I tried but didn't get this..

Kindly help please...

Answer 1

1) Is there any element $r \in R$ with $\varphi(r) = (r,1)$? Is $(r, 1) \in R \times \mathbb{Z}$?

2) What does the image of $\varphi$ look like? Do you by any chance already have a homomorphism $R \to Im(\varphi)$? What does the kernel of this homomorphism look like? Can you apply the first Isomorphism Theorem to prove that they are isomorphic?

Answer 2

Hint:

(1) Can you tell which element in $R$ maps to $(r,1)$?

(2) A homomorphism from a set to its image! What do you think about this statement?