Eh, if R is geometric then so is algebra. The only thing geometric about R is that you can define an absolute value (i.e. it is an ordered ring) which is a prop

Eh, if R is geometric then so is algebra. The only thing geometric about R is that you can define an absolute value (i.e. it is an ordered ring) which is a property it inherits from the fractions. The fractions in turn are a basic consequence of the definition of a field (the fractions have a unique image into all fields). If you've got addition and multiplication then you can get the reals by just adding additive inverse, multiplicative inverses and completeness, none of which are inherently geometric properties. So any geometry must come from the distributive property, because there's not much else it could be.