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 is a ring, then is said to be Baer if for any subset theres is an idempotent such that is the set of annihilator of .
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 is a reduced ring and suppose that there are finitely many idempotents (in its complete ring of quotient) such that is a Baer ring. Then 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 be a real Baer ring and let be the total integral closure of such that is in a natural way a finitely-generated -module. Then is actually a real closed ring and
and if I know that the first Conjecture holds true, then I can even remove the requirement of 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 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 should be real.).
Artin-Schreier Thereom. Let be a field and let be its algebraic closure. Suppose furthermore that and are unequal, but is a finite field extension of . Then has actually characteristic 0. In fact, is a real closed field and