[Paper Review] On Fields With Only Finitely Many Maximal Subrings

Alborz Azarang|arXiv (Cornell University)|Dec 16, 2014

Rings, Modules, and Algebras22 references4 citations

TL;DR

This paper completely characterizes fields with only finitely many maximal subrings, proving they are absolutely algebraic fields with a unique subfield lacking maximal subrings and finite degree over it. The key result establishes equivalence between finite maximal subrings, finite chains of maximal subrings, and finite degree extensions over a subfield with no maximal subrings.

ABSTRACT

Fields with only finitely many maximal subrings are completely determined. We show that such fields are certain absolutely algebraic fields and give some characterization of them. In particular, we show that the following conditions are equivalent for a field $E$: 1. $E$ has only finitely many maximal subrings. 2. $E$ has a subfield $F$ which has no maximal subrings and $[E:F]$ is finite. 3. Every descending chain $\cdots\subset R_2\subset R_1\subset R_0=E$ where each $R_i$ is a maximal subring of $R_{i-1}$, $i\geq 1$, is finite. Moreover, if one of the above equivalent conditions holds, then $F$ is unique and contains all subfields of $E$ which have no maximal subrings. Furthermore, all chains in $(3)$ have the same length, $m$ say, and $R_m=F$, where $m$ is the sum of all powers of primes in the factorization of $[E:F]$ into prime numbers.\\ We also determine when certain affine rings have only finitely many maximal subrings. In particular, we prove that if $R=F[α_1,\ldots,α_n]$ is an affine integral domain over a field $F$, then $R$ has only finitely many maximal subrings if and only if $F$ has only finitely many maximal subrings and each $α_i$ is algebraic over $F$, which is similar to the celebrated Zariski's Lemma. Finally, we show that if $R$ is an uncountable PID then $R$ has at least $|R|$-many maximal subrings.

Motivation & Objective

• To completely classify fields that possess only finitely many maximal subrings.
• To determine the structural conditions under which a field extension has finitely many maximal subrings.
• To characterize when affine integral domains over a field have finitely many maximal subrings.
• To establish a connection between the number of maximal subrings and the algebraic structure of the base field and extension.

Proposed method

• Establishing equivalence between three conditions: (1) finite maximal subrings, (2) finite degree extension over a subfield with no maximal subrings, and (3) finite chains of maximal subrings.
• Using the structure of maximal subrings in integral extensions and properties of absolutely algebraic fields.
• Applying results from Zariski’s Lemma and Artin-Tate Theorem to analyze affine domains and their maximal subrings.
• Employing chain constructions via irreducible elements and units in PIDs to derive lower bounds on the number of maximal subrings.
• Analyzing uncountable PIDs to show that they must have at least |R| many maximal subrings, using algebraic independence and irreducibility arguments.
• Using the fact that if a ring has infinitely many irreducible units, it admits an infinite descending chain of maximal subrings.

Experimental results

Research questions

• RQ1 Which fields have only finitely many maximal subrings?
• RQ2 What structural conditions on a field extension ensure that the larger field has only finitely many maximal subrings?
• RQ3 Under what conditions does an affine integral domain over a field have finitely many maximal subrings?
• RQ4 How does the number of maximal subrings relate to the algebraic structure of the base field and the extension?
• RQ5 Can infinite descending chains of maximal subrings be constructed in uncountable PIDs, and what does this imply about the number of maximal subrings?

Key findings

• A field $ E $ has only finitely many maximal subrings if and only if it is a finite extension of a subfield $ F $ that has no maximal subrings.
• The subfield $ F $ is unique and contains all subfields of $ E $ that have no maximal subrings.
• All chains of maximal subrings in $ E $ have the same finite length $ m $, and $ R_m = F $, where $ m $ is the sum of the powers of primes in the factorization of $ [E:F] $.
• An affine integral domain $ R = F[eta_1, eta_2, ewlineeta_n] $ has finitely many maximal subrings if and only if $ F $ has finitely many maximal subrings and each $ eta_i $ is algebraic over $ F $, analogous to Zariski’s Lemma.
• An uncountable PID $ R $ has at least $ |R| $-many maximal subrings, and such rings admit infinite descending chains of maximal subrings.
• If a field $ K $ has finitely many maximal subrings, then the ring $ T = K[eta] $ with $ eta^2 = 0 $ has $ 1 + |RgMax(K)| $ maximal subrings.