Inverses in the quotient ring $\mathbb{C}[x_1,\dots, x_n]/(x_1^2,\dots, x_n^2)$

Question

Consider the quotient ring $R = \mathbb{C}[x_1,\dots, x_n]/(x_1^2,\dots, x_n^2)$.

Do the elements $1 - (x_1+ \cdots + x_r)$ have explicit inverses in $R$?

For instance, the element $1-x_1$ has inverse $1+x_1$.

Answer 1

The inverse of $1-(x_1+\cdots+x_r)$ is $$\sum_{k=0}^\infty(x_1+\cdots+x_r)^k.$$ An infinite sum, I hear you complain! Not so, from $k=r+1$ onwards, every term is zero inside $R$, and it's really a finite sum.