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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.
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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