ExampleConstruction: Literature-sourcedVerification: Literature-sourcedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-27

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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.
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Dual numbers and their reduced quotient have the same prime set

Example

Let $k$ be a field and let $A = k [ε] / (ε^{2})$ . Then $Spec (A)$ consists of the single prime ideal $(\overline{ε})$ , and the reduced quotient $A / (\overline{ε}) ≅ k$ has the same prime spectrum.

Facts & Assumptions

Given: A field $k$ and the dual-number ring $A = k [ε] / (ε^{2})$ .

[L1]

Prime ideals of a quotient ring correspond to prime ideals of the original ring containing the kernel (Prime ideals of a quotient ring are exactly the prime ideals containing the ideal).

[L2]

Passing to the reduced quotient does not change the prime spectrum (Passing to the reduced quotient does not change the prime spectrum).

Verification

technique · direct

1.1

By [L1], prime ideals of $A$ correspond to prime ideals of $k [ε]$ containing $(ε^{2})$ . Any such prime contains $ε$ because $ε^{2}$ lies in it. The ideal $(ε)$ itself is prime since $k [ε] / (ε) ≅ k$ is a field. Therefore $Spec (A) = {(\overline{ε})}$ .

L1givenalgebra

2.1

The element $\overline{ε} \in A$ is nilpotent, so the reduced quotient of $A$ is exactly $A / (\overline{ε}) ≅ k$ . By [L2], this quotient has the same prime spectrum as $A$ , which is the singleton from step 1.1.

L2step 1.1

3.1

Hence the dual numbers and their reduced quotient have the same prime set.

step 1.1step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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Sources

J. S. Milne, A Primer of Commutative Algebra, v4.03, §14 The spectrum of a ring (standard reference, not scraped)
The Stacks Project, Section 10.17: The spectrum of a ring (standard reference, not scraped)