TITLE: A torsor equivalent for a ring
QUESTION [5 upvotes]: Reading John Baez's essay on torsors, I was quite intrigued with the last section which states:

"Finally, one more remark for people who want to go further. Near the beginning of this essay, I said "as soon as we pick units of temperature, temperatures are elements of an R-torsor". We need to pick units of temperature to know what it means to "add 1" to a temperature. So, where should we think of temperatures a living before we pick units? We should think of them as living on a line whose symmetries include not just translations but also dilations - in other words, the "stretchings" or "squashings" that result from a change of units. Picking an origin reduces the symmetry group to just dilations - and indeed, there's a distinguished choice of origin, namely absolutely zero. Picking units reduces the symmetry group to just translations, giving us an R-torsor - and indeed, there's a distinguished choice of units, namely Planck units. Picking both lets us think of temperatures as real numbers. This combination of translations and dilations arises because R is not just a group, but a ring. So, there's a more sophisticated concept than that of "torsor" allowing both translations and dilations whenever you start with a ring."

So, what is a generalization of a torsor for a ring?

REPLY [4 votes]: A free module of rank one. ${}{}{}$