We know that if an ideal $ I$ of a commutative ring $ R$ with identity, is finitely generated then we can find a minimal generating set for it, that is, a set of elements of $ I$ such that they generate $ I$ and any proper subset of them cannot generates $ I$ . Now let $ I $ be a non finitely generated ideal of a commutative ring with identity. I want to know if there exists a generating set for $ I$ , say $ X$ , and a subset $ X’$ of $ X$ such that neither $ X’$ nor $ X-X’$ is a generating set for $ I $ ?
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