If you view a $\mathbb{Z}$-valued (Cartier) divisor (on say, an integral separated scheme X) as a $\mathbb{G}\_m$-torsor on X with generic trivialization, then for any torus T with character group $X^\ast(T)$, an $X^\ast(T)$-valued divisor on X is a T-torsor with generic trivialization. This shows up for X a curve, e.g., in some treatments of geometric class field theory and Gaitsgory's Twisted Whittaker model paper. The analogous construction will work for any group of multiplicative type, replacing the character group with a suitable fpqc sheaf of abelian groups on the base.

If R is a number ring, you can demand that $X^\ast(T)$ be an R-module, so in this case, you're looking for torsors under tori with CM, with generic trivializations. I don't know where or if this is used, but it seems interesting enough.

If you view a $\mathbb{Z}$-valued Cartier divisor (on say, an integral separated scheme X) as a $\mathbb{G}\_m$-torsor on X with generic trivialization, then for any torus T with character group $X^\ast(T)$, an $X^\ast(T)$-valued divisor on X is a T-torsor with generic trivialization. The analogous construction will work for any group of multiplicative type, if you replace the character group with a suitable fpqc sheaf of abelian groups on the base. This shows up for X a curve in some treatments of geometric class field theory, since the space of these divisors is the affine Grassmannian (in the sense of Beilinson-Drinfeld) for T over the Ran space of X. This space is used in, e.g., Gaitsgory's Twisted Whittaker model paper, where it forms a home (together with some similar objects) for factorizable sheaves.

If R is a number ring, you can demand that $X^\ast(T)$ be a sheaf of R-modules, so in this case, you're looking for torsors under tori with CM, with generic trivializations. I don't know where or if this is used, but it seems interesting enough.

If you view a $\mathbb{Z}$-valued (Cartier) divisor (on say, an integral separated scheme X) as a $\mathbb{G}\_m$-torsor on X with generic trivialization, then for any torus T with character group $X^\ast(T)$, an $X^\ast(T)$-valued divisor on X is a T-torsor with generic trivialization. This shows up for X a curve, e.g., in some treatments of geometric class field theory and Gaitsgory's Twisted Whittaker model paper. The analogous construction will work for any group of multiplicative type, replacing the character group with a suitable fpqc sheaf of abelian groups on the base.

I don't know what you get if you want the action of a larger ring - you can certainly tensor up the character groups as abelian sheaves, so if you're looking for tori with some kind of CM, I guess that will work.

If you view a $\mathbb{Z}$-valued (Cartier) divisor (on say, an integral separated scheme X) as a $\mathbb{G}\_m$-torsor on X with generic trivialization, then for any torus T with character group $X^\ast(T)$, an $X^\ast(T)$-valued divisor on X is a T-torsor with generic trivialization. This shows up for X a curve, e.g., in some treatments of geometric class field theory and Gaitsgory's Twisted Whittaker model paper. The analogous construction will work for any group of multiplicative type, replacing the character group with a suitable fpqc sheaf of abelian groups on the base.

If R is a number ring, you can demand that $X^\ast(T)$ be an R-module, so in this case, you're looking for torsors under tori with CM, with generic trivializations. I don't know where or if this is used, but it seems interesting enough.