quotient of polynomial ring in two variables as a module isomorphic to polynomial ring in one variable?

Question

Let $R=\Bbb C[t]$ be the ring of polynomials in $t$. I am studying the module $M:=R[x]/(x-t)$ as a $R$-module. Is it true that $M$ is isomorphic to $R$ as an $R$-module? It seems to be that the substitution map substituting $t=x$ would give such an isomorphism, but I am not sure if this map is injective.

Answer 1

The evaluation map is not injective, but it does give you what you want.

$$ev_t:\begin{cases} R[x]\to R \\ p(x)\mapsto p(t) \end{cases}$$

is clearly surjective. So by the first isomorphism theorem, $R\cong R[x]/(x-t)=M$ since $(x-t)$ is the kernel of the homomorphism.

Answer 2

For any ring $R$, and any $a\in R$, one has $R[x]/(x-a)\simeq R$. The map is induced by the map $R[x]\to R,\enspace p(x)\mapsto p(a)$, which is onto, and its kernel is the ideal $(x-a)$ of $R[x]$.