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

A minimal primary decomposition is a finite expressionin proper primary ideals such that the prime ideals ${\mathfrak{p}}_j=\sqrt{{\mathfrak{q}}_j}$ are pairwise distinct and no primary ideal can be omitted. More explicitly, for every $j$,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.