In the paper the homological algebra of Artin groups by Craig C. Squire it is stated that the following ring:$$R=\mathbb{Z}[s^{\pm 1},t^{\pm 1}]/(t^2-t+1,(st+1)(s-1))$$seen as an abelian group has rank $4$. It is obvious that, as an abelian group with the sum, it is generated by $1,t,s,st$. Indeed, computing the ranks of quotients of $\mathbb{Z}[s^{\pm 1},t^{\pm 1}]$ is fairly simple. Now, I was trying to pass from $\mathbb{Z}$ to $\mathbb{Q}$ and to compute the dimension over $\mathbb{Q}$ of the following rings$$R_1=\mathbb{Q}[s^{\pm 1},t^{\pm 1}]/(t^2-t+1,(st+1)(s-1))$$$$R_2=\mathbb{Q}[s^{\pm 1},t^{\pm 1}]/(s+1,s^{n-1}t+1)$$Here I'm not sure how to proceed, I don't know if $\dim_\mathbb{Q}R_i$ coincides with the Krull dimension and if there is some easy way to compute these dimensions.
Any help will be highly appreciated.
