Baur-Monk quantiﬁer elimination theorem asserts that any formula in the language of modules is modulo the theory a boolean combination of BG-Invariants and positive primitive formulas. However, in p.54 Model theory of Modules by Prest, just right after proving the theorem, it was immediately concluded as Cor 2.15 that every sentence is a boolean combination of BG-Invariant conditions. Why does it follow? Thanks.



Question on Baur-Monk quantiﬁer elimination for modules
By Baur-Monk QE, every sentence is equivalent to a boolean combination of BG-invariants and positive primitive *sentences* (which have no free variables).


A positive primitive sentence $\psi$ has the form $\exists x\_1\dots\exists x\_n\,\varphi(x\_1,\dots,x\_n)$, where $\varphi$ is a finite conjunction of equations between linear combinations of the variables $x\_1,\dots,x\_n$. Since $\varphi(0,\dots,0)$ is true in every module, $\psi$ is true in every module.


It follows that the Boolean combination of BG-invariants and positive primitive sentences can be simplified to remove the positive primitive sentences.


