Chain Conditions, Semisimple Modules and the Wedderburn–Artin Theorem — Examples

About this page

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

✓ 12 results · all verified · 6 also independently AI-judged

Every result on this page is machine-checked by a proof checker and read in full and owner-audited; the judge is an additional, independent cross-model AI review of the proofs. The 6 not AI-judged were verified by owner audit (typically over a confirmed judge false positive), not failures.

1 · Prerequisites

Abstract Algebra

Combinatorics

Finite Counting, Factorials and Binomial Coefficients

Linear Algebra

Number Theory

Real Analysis

Set Theory

2 · Logical flowchart

3 · Items

Example

Chain Conditions, Semisimple Modules and the Wedderburn–Artin Theorem — Examples

3 · Items

ℤ is Noetherian but not Artinian as a module over itself

The Prüfer p-group is Artinian but not Noetherian

The ring (ℤ/2)^ℕ is not Noetherian

Mₙ(F) as a direct sum of minimal left ideals

A product of two fields is semisimple with two simple-module types

The ℤ-module ℤ/pᵏ has length k

√2 and (1+√5)/2 are algebraic integers, while 1/2 is not

False statement: every Artinian module is Noetherian

False statement: every module has a composition series

False statement: every semisimple ring is commutative

False statement: every subring of a Noetherian ring is Noetherian

False statement: every right Noetherian ring is left Noetherian