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On ideals of rings of fractions and rings of polynomials. (English) Zbl 1331.13002
The set of ideals of a commutative ring $$R$$, denoted by the authors as Idl$$(R)$$, is a distributive lattice under containment. In this paper, the authors consider the relationship between the lattices Idl$$(R)$$ and Idl$$(R^{\prime})$$ for rings $$R^{\prime}$$ which are either localizations of $$R$$ or polynomial extensions of $$R$$. One of the questions they consider is: when is a subset of ideals $$\mathcal{A} \subseteq \text{ Idl}(R)$$ precisely the set of preimages of ideals of $$S^{-1}R$$ for a multiplicatively closed subset $$S$$ of $$R$$? In the case that $$R = \mathbb{Z}$$, they show a subset $$\mathcal{A}$$ of Idl$$(\mathbb{Z})$$ is defined by the preimages of ideals of $$S^{-1}\mathbb{Z}$$ if and only if the set of ideals in $$\mathcal{A}$$ satisfy the property $$IJ \in \mathcal{A}$$ if and only if $$I \in \mathcal{A}$$ and $$J \in \mathcal{A}$$. They also present the notion of preorder embeddings on partially ordered sets and note that if there is a preorder embedding $$g:P \rightarrow Q$$ and $$Q$$ satisfies ACC (or DCC) then $$P$$ satisfies ACC (or DCC). From any poset $$P$$, they define a poset $${\mathbf{N}}(P)$$ to be the set of mappings $$f: \mathbb{Z}_{\geq 0} \rightarrow P$$ with order $$\leq$$ defined by $$f \leq g$$ if and only if $$f(n) \leq g(n)$$ for all $$n \in \mathbb{Z}_{\geq 0}$$. First they prove that $$P$$ satisfies ACC (or DCC) if and only if $${\mathbf{N}}(P)$$ does and then they prove there is a preorder embedding between Idl$$(R[x])$$ and the poset $${\mathbf{N}}(\text{Idl}(R))$$ illustrating a new proof of the Hilbert Basis Theorem.
##### MSC:
 13A15 Ideals and multiplicative ideal theory in commutative rings 06A06 Partial orders, general 13E05 Commutative Noetherian rings and modules 13F20 Polynomial rings and ideals; rings of integer-valued polynomials
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