In the study of quadratic spaces over general rings, there is a type of scalar which people consider called a
Form ring $(R,\Lambda)$ relative to some anti-automorphism denoted $(-)^J:R\to R$ and some $\varepsilon \in R^*$, such that:
- $(-)^{J^2} = \varepsilon(-)\varepsilon^{-1}$,
- $\varepsilon^J = \varepsilon^{-1}$,
- $\Lambda$ is a subgroup of $(R,+)$ satisfying some highly involved conditions which I won't include here.
Its definition is rather involved, as you can see. And when $2 \in R$ is a unit, then $\Lambda$ is fully determined.
When $2 \in R$ is a unit, there is no choice but to get $\Lambda = \{r - r^J\varepsilon \mid r \in R\}$.
Is there any reason to consider the notion of a form ring, and specifically the abelian group $\Lambda$, when $2 \in R$ is a unit?
