Understanding Baer Rings

I have been kind of inactive in mathematics, but hopefully not yet infertile. Recently there is a big thing that has been teasing me and I have to write about it. If not anywhere, at least here!

Here, we will always assume commutative unitary rings.

Let’s start with a simple definition:

Definition: Suppose that A is a ring, then A is said to be Baer if for any subset S\subset A theres is an idempotent e\in A such that eA is the set of annihilator of S.

It was in the 70’s that Mewborn showed that every reduced commutative ring has a Baer Hull, this is the smallest intermediate ring between the ring and it’s complete ring of quotients that is Baer. Mewborn showed a more detailed construction of the Baer hull of a reduced commutative ring. He showed that the Baer Hull of a reduced commutative ring is just the adjoint of the ring with all the idempotents of its complete ring of quotients.

But… there is a question, rather a conjecture, that lingers in my head…

Conjecture. Suppose that A is a reduced ring and suppose that there are finitely many idempotents (in its complete ring of quotient) e_1,e_2,\dots, e_n such that A[e_1,\dots,e_n] is a Baer ring. Then A is actually a Baer ring.

I’m thinking about this for a while, hope that the answer pops in my head soon. But why am I trying so hard to show this? Well…

I think I can show/prove the following:

Conjecture. Let A be a real Baer ring and let T be the total integral closure of A such that T is in a natural way a finitely-generated A-module. Then A is actually a real closed ring and

T= A[\sqrt{-1}]

and if I know that the first Conjecture holds true, then I can even remove the requirement of A being Baer. This would be great!

But, why am I trying so hard to show that this second Conjecture holds?? Simply because the second Conjecture is beautiful 🙂 not to mention the fact that it is a generalization of the “classical” Artin Schreier Theorem (if you A were a field, then its total integral closure is no other than its algebraic closure. As a note, I am afraid I cannot remove the requirement that A should be real.).

Artin-Schreier Thereom. Let F be a field and let K be its algebraic closure. Suppose furthermore that F and K are unequal, but K is a finite field extension of F. Then F has actually characteristic 0. In fact, F is a real closed field and

K = F[\sqrt{-1}]

Ancel C. Mewborn, “Regular Rings and Baer Rings”, Math. Z. 1971, vol. 121, p. 211-219

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