A semialgebraic closure for commutative algebra

Let A⊂RA⊂R be rings containing the rationals. In RR let SS be a multiplicatively closed subset such that 1∈S1∈S and 0∉S0∉S, TT a preorder of RR (a proper subsemiring containing the squares) such that S⊂TS⊂T and II an AA-submodule of RR. Define ρ(I)ρ(I) (or ρS,T(I)ρS,T(I)) to be ρ(I)={a∈R|sa2m+t∈I2m for some m∈N,s∈S and t∈T}.ρ(I)={a∈R|sa2m+t∈I2m for some m∈N,s∈S and t∈T}. We show that ρρ is a closure operator on AA-submodules, establish some of its properties, and motivate its introduction by considering the assignment of characteristically real multiplicities to points of real varieties and semialgebraic sets.