What is the required homomorphism satisfying $f(c)=c$ for all $c\in R$ and $f(X)=aX+b$?

Question

My question is related to this post.

I know from the Proposition that

Let $φ : R → R'$ be a ring homomorphism. Given elements $a_1, · · · , a_n ∈ R'$ , there is a unique homomorphism $Φ : R[x_1, · · · , x_n] → R'$ , which agrees with φ on constant polynomials, and which sends $x_i$ to $a_i$ f0r each $i$.

The desired homomorphism is given by $Φ(\sum_{i=0}^nc_ix^i) =\sum_{i=0}^n φ(c_i)a^i$ for $c_i ∈ R$.

I need to know what is analogous homomorphism for the problem in the post.

Thanks

Answer 1

Since $f(c)=c$ for all $c$ then

for $c=0$ we get $a0+b=f(0) =0$ so $b=0$ and

for $c=1$ we get $a1 +0 = f(1) =1$ so $a=1$.

Thus $f(X) =X$.