Math and science::Algebra::Aluffi

Group isomorphism: quotient of R[x] ≅ direct sum of R

In what way can we construct an isomorphism using $R [x]$ and $R$ ?

If we restrict our aim to just a group isomorphism then we can achieve the following:

Let $R$ be a commutative ring, $R [x]$ a polynomial ring, and $f (x) \in R [x]$ a polynomial. Let [ $φ : R [x] \to ?$ ] be a set function defined by sending $g (x) \in R [x]$ to the tuple representation of the remainder of $g (x)$ when divided by $f (x)$ .

Then this function induces an isomorphism of abelian groups:

[

? ≅ ?

]

Can you construct the proof?

Extra:

How to construct the complex ring $C$ ?
How to construct a ring of polynomial evaluations?

11.07.2023