Let $k$ a field, $A = k[x,y]/(y^2-x^3)$ and $\mathfrak{a} = (x,y)$ and let $G_{\mathfrak{a}}(A) := \bigoplus_{n \geq 0} \mathfrak{a}^n/\mathfrak{a}^{n+1}$ be the associated graded ring of $A$ with respect to $\mathfrak{a}$.
For each $n$, $\mathfrak{a}^n$ is generated by the monomials $x^n, x^{n-1}y, \dots, y^n$ as a $k$-module, thus generated by $x$ and $y$ as a $k$-algebra. Thus can we then say that $G_{\mathfrak{a}}(A)$ is generated by $x$ and $y$ as a $k$-algebra?
I want to define a graded $k$-algebra morphism $\varphi : G_{\mathfrak{a}}(A) \to k[s,t]/(t^2)$ by sending $x \mapsto s$ and $y \mapsto t$ and then extending the map polynomially from the generators to the whole $k$-algebra, but I am wondering if I am forgetting to check something or made a mistake somewhere.