Is the following ring in integral domain?
Since Ore extensions of domains are Ore extensions, this question can be reformulated as follows: Since the relation look almost like those of a skew polynomial ring, can this ring in fact be written as a skew polynomial ring?
Consider the $k$-algebra $R=k\langle x_i, c_i\rangle_{i=1,\ldots,n}$, $k$ not of characteristic 2, subject to the following relations:
- $x_ix_j = x_jx_i$ for all $i, j$
- $x_ic_i = -c_i x_i$
- $x_ic_j = c_j x_i$ for all $i\neq j$
- $c_ic_j = -c_j c_i$ for all $i\neq j$
I haven't found a counterexample yet.
Indeed, @DietrichBurde's comment turned out to be helpful: Just start with $\mathbf C[x_i]$, take the automorphism $\sigma_1: x_1\mapsto -x_1$ that leaves all other $x_i$ unchanged. Then we can form the skew polynomial ring $\mathbf C[x_i, c_1]$ such that $x_1c_1 = -c_1x_1$. Continue inductively. Hence the ring is a skew polynomial ring and hence remains a domain.