I am trying a problem that asks if it is true that $R[x]/(g) \cong R[x]^{(n-1)}$ (as groups under addition), when $g$ is a monic polynomial of degree $n$, and where $(g)$ is the ideal generated by $g$. Note, $R[x]^{(n)} = \{f \in R[x] : \deg(f) \leq n\}$ and $R$ is a commutative ring.
I know if $g = x^n$, this statement holds intuitively. Though, if $g = x^n + x + 1$ or any other polynomial with more than one term, I'm not so sure.
My hunch is that it is true and I would like to try and apply the first isomorphism theorem, but I cannot find a group homomorphism with kernel $(g)$, given that I believe $(g)$ can be irreducible.
I would appreciate any help on this!
The map $R[x]^{(n-1)}\to R[x]/(g)$ that is induced by the inclusion is injective and onto.