Is $\mathbb{C}[q,q^{-1}]/(q-1)\mathbb{C}[q,q^{-1}] = \mathbb{C}$?

Question

Let $\mathbb{C}(q)$ be the field of rational functions in $q$. Let $\mathbb{C}[q,q^{-1}]$ be the subring of $\mathbb{C}(q)$ consisting of all Laurent polynomials in $q$.

Do we have Is $\mathbb{C}[q,q^{-1}]/((q-1)\mathbb{C}[q,q^{-1}]) = \mathbb{C}$?

Thank you very much.

Answer 1

Note that $\mathbb{C}[q,q^{-1}]/(q-1)$ is isomorphic to $\mathbb{C}[q,t]/(qt-1,q-1)$. Notice that $(qt-1,q-1)=(q-1,t-1)$ which is a maximal ideal.