A real Nullstellensatz with multiplicity

Let A be a commutative ring containing the rationals. Let S be a multiplicatively closed subset such that 1∈S and 0∉S, T a cone in A such that S⊂T and I an ideal in A. ThenρS,TI={a|sa2m+t∈I2mfor some m∈N,s∈S and t∈T} is an ideal. For a commutative ring the collection of non-reduced orders (total cones) is a fibration of the real spectrum. Both concepts carry information regarding multiple solutions in the constructible set associated with I,T and S. When the ring is a real regular domain, a non-reduced Nullstellensatz is presented that extends the real Nullstellensatz and relates these concepts. The notion of real multiplicity is proposed and examined for elements that are either positive definite (PD) or positive semi-definite (PSD) on the real spectrum.