Let $k$ be a field and $k[[t]]$ the ring of formal power series over $k$. A text that I am reading says:
Consider the coordinate ring $k[x,y,t]/(x^2+txy)$ with its canonical $k[[t]]$-module structure.
What is the canonical $k[[t]]$-module structure on $k[x,y,t]/(x^2+txy)$? My guess would be $\sum_i a_i t^i.[\sum_l(\sum_{j,k} b_{i,j,k}x^jy^k)t^l]:=[\sum_l(\sum_{j=0}^l a_jc_{l-j})t^l]$, where $c_l=\sum_{j,k} b_{i,j,k}x^jy^k$. But I am not seeing that this expression even makes sense, i.e. that the formal power series $\sum_l(\sum_{j=0}^l a_jc_{l-j})t^l$ has a representative (w.r.t. modding out by the ideal $(x^2+txy)$) in $k[x,y,t]$.
The text I am reading is based on this paper, namely Exercise 3.14 on page 12. See comments below.