Levitzky's theorem¶
In a right Noetherian ring, every nil left or right ideal has one power that vanishes as an ideal.
Core Idea¶
Levitzky's theorem states that in a right Noetherian ring, every nil left or right ideal is nilpotent. Nil means each member x has some power x^m=0, perhaps with a different m for each member. Nilpotent means there is one n for which the entire ideal power I^n=0; every product of n ideal members vanishes. The ring's Noetherian chain condition makes the stronger, shared-power conclusion possible.[^ref-43447c051fff]
Scope of Application¶
The theorem applies to a left or right ideal when the ambient ring is right Noetherian and every element of the ideal is nilpotent. It need not be the full nilradical or a two-sided ideal. A separate ring-theory source states the side-reversed left-Noetherian form. Neither source permits dropping the chain condition.[ref-43447c051fff][ref-092e47225433]
The worked examples below are finite or finite-dimensional rings that are Artinian as well as Noetherian. They illustrate the conclusion, but do not alone show the theorem's full Noetherian scope or a proper one-sided case.[^ref-092e47225433]
Clarity¶
The key distinction is every x in I has some x^m(x)=0 versus one n makes I^n=0. The second claim concerns mixed products of ideal members, not just repetitions of an individual element. The theorem is also distinct from the nilradical as an object and from the Artinian-starting Hopkins–Levitzki theorem.[^ref-43447c051fff]
Manages Complexity¶
Check three things in order: the ring and its right Noetherian condition, a particular nil left or right ideal, and the conclusion that one ideal power is zero. In Z/200Z, the ideal (10) has nonzero square (100) but zero cube (1000)=(0) modulo 200. In upper-triangular 3×3 complex matrices, the strictly upper-triangular ideal likewise has nonzero square and zero cube. The exponent three is specific to these calculations; the theorem states existence, not a universal value.[^ref-092e47225433]
Abstract Reasoning¶
Once right Noetherianity and nilness of the chosen one-sided ideal have been established, infer that some I^n vanishes. Do not announce the smallest n without a separate calculation. If the ring's chain condition or the ideal's nilness has not been shown, the implication cannot be applied. The converse, from I^n=0 to each x^n=0, is immediate and needs no Noetherian hypothesis.[^ref-43447c051fff]
Knowledge Transfer¶
The same ring-theoretic implication works in commutative and noncommutative eligible rings, but the exponent and ideal structure must be checked in each case. A general analogy from individual behavior to a shared bound does not make this named theorem a Prime: its literal terms require ring multiplication, a one-sided ideal, and the Noetherian condition.[ref-43447c051fff][ref-092e47225433]
Example¶
The Manchester notes identify the strictly upper-triangular ideal I of U_3(C) as its nilradical. This finite-dimensional noncommutative algebra is Noetherian. E12 E23=E13 shows I^2 can be nonzero, while every product of three members of I is zero. Thus the ambient ring, nil one-sided ideal, and one uniform vanishing power are all explicit. This two-sided Artinian example does not limit the theorem to radical ideals or exponent three.[^ref-092e47225433]
Relationships to Other Abstractions¶
Current abstraction Levitzky's theorem Domain-specific
Parents (1) — more general patterns this builds on
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Levitzky's theorem presupposes Ring Ideal Domain-specific
The theorem's nil premise and nilpotent conclusion both concern a one-sided ring ideal.
Hierarchy paths (6) — routes to 5 parentless roots
- Levitzky's theorem → Ring Ideal → Set and Membership
- Levitzky's theorem → Ring Ideal → Ring → Group → Monoid → Identity Element
- Levitzky's theorem → Ring Ideal → Ring → Group → Monoid → Semigroup → Closure
- Levitzky's theorem → Ring Ideal → Ring → Group → Monoid → Semigroup → Set and Membership
- Levitzky's theorem → Ring Ideal → Ring → Group → Monoid → Semigroup → Associativity → Invariance
- Levitzky's theorem → Ring Ideal → Ring → Group → Monoid → Semigroup → Associativity → Symmetry
Neighborhood in Abstraction Space¶
Levitzky's theorem sits in a sparse region of the domain-specific corpus (84th percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.
Family — Algebraic Substructures & Closures (12 abstractions)
Nearest neighbors
- Maximal Ideal — 0.86
- Principal Ideal — 0.83
- Artinian Ideal — 0.82
- V-Ring (Ring Theory) — 0.80
- Initial and terminal objects — 0.80
Computed from structural-signature embeddings · 2026-10-08
Not to Be Confused With¶
A nil ideal has individually nilpotent elements; a nilpotent ideal has a shared zero ideal power. The nilradical is an ideal object, not the theorem itself. Hopkins–Levitzki begins from an Artinian condition and is a distinct result. The left-Noetherian formulation is a separately sourced mirror of the stated right-Noetherian version, not an excuse to omit side conventions.[ref-43447c051fff][ref-092e47225433]
References¶
[^ref-43447c051fff]: L. W. Small, Noncommutative Ring Theory Notes, Definition 3.1 and Theorems 4.2, 4.9–4.10, especially Theorem 4.10 on PDF p. 16 (zero-index p. 15). University of California, San Diego course notes. https://mathweb.ucsd.edu/~lwsmall/MATH207A/ringtheorynotes.pdf
[^ref-092e47225433]: T. Stafford, notes modified by M. Prest, Noncommutative Algebra (2019), Example 4.9 on PDF p. 52 (zero-index p. 51) and Theorem 4.10 on PDF p. 53 (zero-index p. 52). University of Manchester course notes. https://personalpages.manchester.ac.uk/staff/mike.prest/NCAlgCrseNotes2019.pdf