Abstract Factorization Theorems with Applications to Idempotent Factorizations
Laura Cossu, Salvatore Tringali

TL;DR
This paper develops abstract factorization theorems in monoids with a preorder, applying them to rings to improve classical results on idempotent factorizations in endomorphism rings.
Contribution
It introduces new factorization theorems based on $ extpreceq$-artinian properties and applies them to rings, extending and refining classical idempotent factorization results.
Findings
Establishes factorization of elements through $ extpreceq$-irreducibles and $ extpreceq$-quarks.
Provides bounds on the length of factorizations in terms of $ extpreceq$-height.
Improves classical theorems on idempotent factorizations in endomorphism rings.
Abstract
Let be a preorder on a monoid and be an integer . The -height of an is the sup of the integers for which there is a (strictly) -decreasing sequence of -non-units of with (with ), where is a -unit if and a -non-unit otherwise. We say is -artinian if there exists no -decreasing sequence of elements of ; and strongly -artinian if the -height of each element is finite. We establish that, if is -artinian, then each -non-unit factors through the -irreducibles of degree , where a -irreducible of degree is a -non-unit that cannot be written as a product of or fewer -non-units…
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
TopicsRings, Modules, and Algebras
