[Paper Review] Binary modules and their endomorphisms

Hans Schoutens|arXiv (Cornell University)|Dec 10, 2012

Rings, Modules, and Algebras5 references3 citations

TL;DR

This paper introduces binary modules—finitely generated modules over Noetherian rings whose fundamental cycle has only 0 or 1 coefficients—and shows that their endomorphism ring has a commutative quotient after factoring out nilpotent endomorphisms. The key result is that for binary modules without embedded primes, the endomorphism ring is commutative, and such modules embed as ideals in reduced rings.

ABSTRACT

Based upon properties of ordinal length, we introduce a new class of modules, the binary modules, and study their endomorphism ring. The nilpotent endomorphisms form a two-sided ideal, and after factoring this out, we get a commutative ring. In particular, any binary module without embedded primes is isomorphic to an ideal in a reduced ring.

Motivation & Objective

• To define and study binary modules using ordinal length and fundamental cycle properties.
• To analyze the structure of the endomorphism ring of such modules, especially the role of nilpotent endomorphisms.
• To characterize binary modules without embedded primes as ideals in reduced rings via commutative endomorphism rings.
• To generalize the univalent case (valence 1) to higher valence binary modules.
• To establish top-compressibility and structural properties for binary modules with or without embedded primes.

Proposed method

• Define binary modules via the Cantor normal form of their ordinal length, where all coefficients are 0 or 1.
• Use the fundamental cycle of a module, linked to its zeroth local cohomology, to define the valence and binary property.
• Apply semi-additivity of ordinal length and cohomological characterizations to derive structural results.
• Prove that nilpotent endomorphisms form a two-sided ideal in the endomorphism ring of a binary module.
• Factor out the nilpotent ideal to obtain a commutative quotient ring, using induction on the length of the associated prime poset.
• Use Vasconcelos' theorem to embed faithful binary modules without embedded primes as ideals in reduced rings.

Experimental results

Research questions

• RQ1 What structural properties emerge in modules whose fundamental cycle has only coefficients 0 or 1?
• RQ2 How does the endomorphism ring of a binary module behave, particularly regarding nilpotent elements?
• RQ3 Under what conditions is the endomorphism ring of a binary module commutative?
• RQ4 Can binary modules without embedded primes be embedded as ideals in reduced rings?
• RQ5 What is the relationship between top-compressibility and the absence of embedded primes in binary modules?

Key findings

• The set of nilpotent endomorphisms in a binary module forms a two-sided ideal in the endomorphism ring.
• After quotienting by the nilpotent ideal, the endomorphism ring becomes commutative.
• For a binary module without embedded primes, the endomorphism ring is commutative, and the module embeds as an ideal in a reduced ring.
• The nilpotency index of an endomorphism in a binary module is bounded by the length of the poset of associated primes.
• A binary module without embedded primes is top-compressible, meaning it embeds into any of its open submodules.
• A binary ring is top-compressible if and only if it has no embedded primes.