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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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