ExampleConstruction: AI-generatedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-27

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A common nilpotence exponent in a Noetherian quotient

Example

Let $A = k [x, y] / (x^{3}, x^{2} y, y^{4})$ . Then the nilradical of $A$ is $(\overline{x}, \overline{y})$ , and a common nilpotence exponent is $5$ : $(\overline{x}, \overline{y})^{5} = (0)$ .

Facts & Assumptions

Given: A field $k$ and the quotient ring $A = k [x, y] / (x^{3}, x^{2} y, y^{4})$ .

[L1]

In a Noetherian ring the nilradical is a nilpotent ideal (The nilradical of a Noetherian ring is nilpotent).

Verification

technique · direct

1.1

Every element of $(\overline{x}, \overline{y})$ is a $k$ -linear combination of positive-degree residue classes, and every such monomial is nilpotent because powers eventually hit one of the relations $x^{3} = 0$ , $x^{2} y = 0$ , or $y^{4} = 0$ . Thus $(\overline{x}, \overline{y})$ is the nilradical.

givenalgebra

2.1

Any monomial of total degree $5$ in $\overline{x}$ and $\overline{y}$ either has $\overline{y}$ -exponent at least $4$ or $\overline{x}$ -exponent at least $2$ together with a positive $\overline{y}$ -exponent, or else $\overline{x}$ -exponent at least $3$ ; each case is zero in $A$ . Hence $(\overline{x}, \overline{y})^{5} = (0)$ . The theorem [L1] guarantees that some common exponent must exist; this computation shows that $5$ works in this example.

L1step 1.1algebra

3.1

Therefore the nilradical of this Noetherian quotient has a concrete common nilpotence exponent.

step 1.1step 2.1∎

Depends on

The nilradical of a Noetherian ring is nilpotent

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

3 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

M. Hochster, Introduction to Commutative Algebra, Math 614 notes (2020) (standard reference, not scraped)
The Stacks Project, Section 10.32: Rings and modules with finiteness conditions (standard reference, not scraped)