author = {Jose Capco},
title = {{Generalizing Some Results in Field Theory for Rings}},
language = {english},
abstract = {I start this talk by giving some motivation on how the concept of \emph{algebraic extensions} for fields can be generalized for rings (in particular reduced rings). So, we give the definition of generalized algebraic extensions and algebraic closure of reduced rings (the study of these extensions started in the 50's). The definition will require understanding of several types of ring extensions which we shall give including several examples. These examples will show us why some necessary adjustments are required to define a generalized algebraic extension for reduced rings. And finally we are able to define some concepts used for fields e.g. splitting ring of a polynomial, automorphism groups of ring extensions etc. There are two main results that I will present the audience. The first result is about some interesting facts about the splitting rings and their group of automorphisms (fixing the base ring): These rings (unlike in fields) will not necessarily induce a finite group of automorphisms and these groups may not even be finitely generated. For the second result I will define real closed rings (similar to real closed fields) and give a generalized form of Artin-Schreier Theorem: If the algebraic closure of a Baer reduced normal real ring is a finitely generated module (over the given Baer ring) then the ring is real closed and if the ring is von Neumann regular then adjoining it with $\sqrt{-1}$ will give us the algebraic closure of the ring. },
year = {2016},
month = {17-28.10.2016},
note = {Contributed talk at Second International Workshop and Conference on Commutative Algebra (SIWCCA) },
conferencename = {Second International Workshop and Conference on Commutative Algebra (SIWCCA) },
url = {http://www.cdmathtu.edu.np/index.php?show=commutative_algebra}