Generalizing Some Results in Field Theory for Rings

In this paper we introduce some Galois-like theory for commutative reduced Baer rings. We show that the splitting ring of a polynomial over a Bear reduced ring is a finitely generated module. These rings will not always induce a finitely generated group of automorphisms, but the group will be a torsion group with finite exponent. Finally we show a generalized Artin-Schreier theorem: if the algebraic closure of a von Neumann, Baer reduced normal real ring is a finitely generated module then the ring is real closed and adjoining the base ring with √ −1 will even give us its algebraic closure