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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 [\( \varphi: R[x] \to \;\; ? \; \) ] 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:

[\[ \;?\;\; \cong \;\; ? \; \] ]

Can you construct the proof?

Extra:

  • How to construct the complex ring \( \mathbb{C} \)?
  • How to construct a ring of polynomial evaluations?