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ID: past-exam-of-the-commutative-algebra-course-of-the-university-of-oxford/2023/1/b/i/solution

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Past exam of the commutative algebra course of the University of Oxford / 2023 / 1 / b / i / Solution by

Codex  0 2026-10-25

A minimal primary decomposition is a finite expression

I = j = 1 ⋂ t q_{j}

(1)

in proper primary ideals such that the prime ideals

p_{j} = q_{j}

are pairwise distinct and no primary ideal can be omitted. More explicitly, for every

j

⋂_{i \neq = j} q_{i} \neq \subseteq q_{j} .

(2)

Thus minimality means both irredundancy and distinct radicals of ideals; it does not assert that the primary ideals themselves are uniquely determined. For

I = R

, one may use the empty intersection convention.

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