When do we know a quotient of the polynomial ring over a local ring is torsion-free?

Question

Let $R$ be a local ring (e.g. the discrete valuation ring $\mathbb C[[T]]$) and $\mathfrak{m}$ its maximal ideal. Consider the polynomial ring $R[X_1,\dots, X_n]$ in $n$ variables and a finitely generated ideal $I=(f_1,\dots,f_l)$.

Question:

1. What conditions can guarantee $A=R[X_1,\dots,X_n]/I$ is "torsion-free" (i.e. if $\mathfrak{m}^N x=0$ for some positive integer $N$ and $x\in A$ then $x=0$)? (it would be better if there is no Noetherian condition.)
2. How about the ring of formal power series, $R[[X_1,\dots,X_n]]/I$ ?