Köthe conjecture

The open ring-theoretic conjecture that the sum of two nil left ideals is nil, equivalently that a ring with no nonzero nil ideal has no nonzero nil one-sided ideal.

Version

v1 · 2026-09-08 · History

Domain-specific #

5222

Origin domain

ring theory

Subdomain

ring theory

Core Idea

Several equivalent formulations connect nil one-sided ideals, the upper nilradical, matrix rings and polynomial-ring Jacobson radicals; the conjecture is established for important ring classes but remains open in general. Starting from a nil one-sided ideal, the equivalences translate closure under sums into containment in the upper nilradical and stability of nil ideals under matrix or polynomial constructions. The abstraction is therefore identified by a declared carrier, a transformation or constraint over that carrier, and an invariant that tells an analyst whether the named structure is genuinely present.

Scope of Application

Köthe conjecture belongs to ring theory and is useful where the analyst can specify the typed ring theory carrier, defining objects and relations, parameters, conventions, evidence, boundary cases, and comparison targets, then evaluate the ambient ring convention, nil versus nilpotent condition, sidedness, chosen equivalent formulation, matrix or polynomial construction, ring-class hypotheses, and open-status claim are explicit. The scope is broad within that domain but bounded by the need for the ambient ring convention, nil versus nilpotent condition, sidedness, chosen equivalent formulation, matrix or polynomial construction, ring-class hypotheses, and open-status claim are explicit. The entry records a descriptive analytical identity; practical use requires the governing domain's evidence, standards, and safety obligations.

Clarity

The abstraction clarifies a crowded vocabulary by making the ambient ring convention, nil versus nilpotent condition, sidedness, chosen equivalent formulation, matrix or polynomial construction, ring-class hypotheses, and open-status claim are explicit the center of the account. A claim should name the carrier, the governing operation or relation, the applicable assumptions, and the recognition test. A bare label is insufficient because the name Köthe conjecture can be used for a formal identity, an implementation, or a neighboring result unless carrier and convention are stated.

Manages Complexity

Without the abstraction, an analyst must reason directly over many local details: the carrier roles, admissibility assumptions, competing conventions, derived invariants, boundary cases, and proof or validation obligations specific to Köthe conjecture. Köthe conjecture compresses them into the roles in the structural signature. That compression permits comparison across instances without erasing the variables that determine validity. It also exposes which details may be varied safely and which are constitutive.

Abstract Reasoning

1. Identify the carrier. State what the elements, states, objects, or observations are: the typed ring theory carrier, defining objects and relations, parameters, conventions, evidence, boundary cases, and comparison targets. Reject examples whose alleged carrier belongs to a different problem. 2. Lock the constitutive rule. Express the ambient ring convention, nil versus nilpotent condition, sidedness, chosen equivalent formulation, matrix or polynomial construction, ring-class hypotheses, and open-status claim are explicit independently of one notation or implementation.

Knowledge Transfer

Knowledge transfers strongly among subfields of ring theory because they reuse the typed ring theory carrier, defining objects and relations, parameters, conventions, evidence, boundary cases, and comparison targets, Starting from a nil one-sided ideal, the equivalences translate closure under sums into containment in the upper nilradical and stability of nil ideals under matrix or polynomial constructions., and type the carrier, state every parameter and convention in the definition, test that the ambient ring convention, nil versus nilpotent condition, sidedness, chosen equivalent formulation, matrix or polynomial construction, ring-class hypotheses, and open-status claim are explicit, compare the nearest accepted identity, and report counterexamples, uncertainty, and limiting cases.

Relationships to Other Abstractions

Current abstraction Köthe conjecture Domain-specific

Parents (1) — more general patterns this builds on

• Köthe conjecture is a kind of Equivalence-Preserving Rewriting Prime

  The proposed strict upward parent is prime:equivalence_preserving_rewriting.

Hierarchy paths (2) — routes to 2 parentless roots

• Köthe conjecture → Equivalence-Preserving Rewriting → Transformation → Function (Mapping)

Show alternative path (1)

Neighborhood in Abstraction Space

Köthe conjecture sits in a crowded region of the domain-specific corpus (27^th percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.

Family — Ring Structure & Module Theory (18 abstractions)

Nearest neighbors

• Primitive ring — 0.92
• Polynomial identity ring — 0.92
• Domain (ring theory) — 0.91
• Gelfand ring — 0.91
• Depth (ring theory) — 0.91

Computed from structural-signature embeddings · 2026-09-08