Category Archives: Ring Theory

Generalizing Galois Theory for Commutative Rings - Part I

I am not sure why this idea has lost popularity after the 60’s. Papers appear about this, but more people seemed to be interested in it half a century ago. I would say, the idea started independently. On one hand we had a group of noncommutative algebraist and homological algebraist working on ideas that was probably once inspired from the category of fields. For instance separable algebras (separable fields) and von Neumann regular rings with its semiheriditary and quasi-inverse property (not far related to fields and product of fields) were, in my opinion, quit popular among noncommutative algebraist and homological algebraist. Then there were a group of people who purposely wanted to see ideas developed for Galois theory extended to rings. We have now definitions for separable ring extensions, splitting ring extensions and even algebraic extensions of rings (which is not the usual algebraic extension we would intuitively define). The last topic (algebraic extension) was studied by Borho, Enochs, Hochster and Raphael.

Having said that, I decided to add myself into the set of cooks (to make a better broth). Recently I proved the following for instance:

Proposition Let $A$ be a Baer ring and $B$ be its total integral closure (this is also called algebraic closure by Robert Raphael) and suppose $f\in A[x]$ is a non-zero monic polynomial over $A$. Consider the set of zeros $S$ of $f$ in $B$. Then $A[S]$ is a finitely generated module over $A$.

The proof is a bit technical and to share it I am going to give a lecture about it and write an article about this (to be continued…).

Edit: I have a lot of new results here but I decided not to write a second post about this yet. I think a pdf file is better for this kind of thing. My paper related to this topic and proofs can be found here.

Generalizing Galois Groups

$\newcommand{\N}{\mathbb{N}}
\newcommand\Q{\mathbb{Q}}
$
There were many attempt to generalize the notion of algebraic extension of fields to other (more general) categories. One of my favourite generalization is for the category of reduced commutative rings which was made popular by the likes of Edgar Enochs, Robert Raphael and Mel Hochster. An algebraic extension in this category is just an essential extension that is an integral extension. Why is integral extension alone not enough? One simple reason is because one can never end an integral extension in this category (you can always find a strict integral extension of a reduced commutative unitary ring that is also reduced and commutative). The necessity of essential extension (essential extensions can be defined in a pure category theoretical way) allows a “largest” algebraic closure. In fact, Hochster has shown that any such reduced commutative unitary ring $A$ will have a largest essential and integral extension which is called the total integral closure of the ring. By largest we mean that for any essential and integral extension of $A$ there is an $A$-monomorphism from this extension to the total integral closure.

The total integral closure is also rightfully known as the algebraic closure of the ring. This name is justified considering the following characterization (made by Hochster):

Let $B$ be the total integral closure of $A$ then.
– All monic polynomials of degree $n\in\N$ with coefficients in $A$ are factored into $n$ linear polynomials with coefficients in $B$
– All residue domains with respect to ideals of $B$ are integrally closed in their algebraically closed field of fractions
(specifically all residue fields with respect to maximal ideals are algebraically closed)

This easily leads to the characterization of algebraically closed domains:
A domain is algebraically closed iff it is integrally closed and if its field of fraction is algebraically closed.

More was investigated by Raphael in the 90s who mostly looked at the von Neumann regular rings that are algebraically closed.

The next question one could pose is the following:
The fundamental groups in Galois theory enjoys the benefit of being finite. Can this be true for $A$-monomorphisms between an essential and integral extension of $A$ and its algebraic closure? I will give example for which we get infinitely such $A$-monomorphisms:

Let $A = \Q^\N$, then $\Q$ itself can be canonically be embedded (as a subring) of $A$ (namely taking the sequence for which all elements are equal). Then the polynomial $f:=(x^2-2)(x^2-3)$ is in $A[x]$ (we clearly abused notation here, $2$ (resp. $3$) is just the sequence of repeating $2$ (resp. $3$)). This polynomial has infinitely many zeros in the overring $B:=\Q(\sqrt{2},\sqrt{3})^\N$ of $A$ and clearly $B$ is both essential and integral extension of $A$. There are therefore infinitely many such zeros that can be mapped onto each other (product of the maps obtained from the usual Galois groups).

I do however believe that if we work with only one polynomial say $f\in A[x]$ then extending $A$ within $B$ such that it contains all the zeros of $f$ will give me a finitely generated $A$-module if $A$ is a Baer reduced commutative unitary ring. I will give a more detailed discussion about this in a next blog.

[1] E. Enochs, Totally Integrally Closed Rings. Proc. Amer. Math. Soc. 1968, Vol. 19, No. 3, p. 701-706.
[2] M. Hochster, Totally Integrally Closed Rings and Extremal Spaces. Pac. J. Math. 1969, Vol. 142, p. 767-779.
[3] R.M. Raphael, Algebraic Extensions of Commutative Regular Rings. Canad. J. Math. 1970, Vol. 22, p. 1133-1155.