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Some extensions of rings with Noetherian spectrum. (English) Zbl 1493.13009

Let \(R\) be a commutative ring with identity. Recall that an ideal \(I\) of \(R\) is called radically finite if there exists a finitely generated ideal \(J\) of \(R\) such that \(\sqrt{I}=\sqrt{J};\) and \(R\) is said to have Noetherian spectrum if each ideal of \(R\) is radically finite. It is easy to see that an ideal \(I\) of \(R\) is radically finite if and only if \(\sqrt{I}=\sqrt{J}\) for some finitely generated subideal \(J\) of \(I\).
In the paper under review, the authors study rings with Noetherian spectrum, rings with locally Noetherian spectrum (a commutative ring \(R\) is said to have locally Noetherian spectrum if \(D_M\) has Noetherian spectrum for all maximal ideals \(M\) of \(R\)) and rings with \(t\)-locally Noetherian spectrum (We say that an integral domain \(D\) has \(t\)-locally Noetherian spectrum if \(D_M\) has Noetherian spectrum for all maximal \(t\)-ideals \(M\) of \(D)\) in terms of the polynomial ring, the Serre’s conjecture ring, the Nagata ring and the \(t\)-Nagata ring. First, let us recall the following notions.
Let \(R\) be a commutative ring with identity and let \(R[X]\) be the polynomial ring over \(R\). Let \(U\) be the set of monic polynomials in \(R[X]\). Then \(U\) is a multiplicative subset of \(R[X]\) and the quotient ring \(R[X]_U\) is called the Serre’s conjecture ring of \(R\). For an element \(f \in R[X]\), \(c(f)\) denotes the content ideal of \(f\), i.e., the ideal of \(R\) generated by the coefficients of \(f\). Let \(N = \{f\in R[X]\mid c(f)=R\}\). Then it was shown that \(N = R[X] \backslash \cup_{M\in Max(R)}MR[X]\) and \(N\) is a saturated multiplicative subset of \(R[X]\) consisting of regular elements of \(R[X]\) [M. Nagata, Local rings. New York and London: Interscience Publishers, a division of John Wiley and Sons (1962; Zbl 0123.03402), page 17 and 18]. The quotient ring \(R[X]_N\) is called the Nagata ring of \(R.\)
The authors proved the following three statements.
1.
A commutative ring \(R\) with identity has Noetherian spectrum if and only if the Serre’s conjecture ring \(R[X]_U\) has Noetherian spectrum, if and only if the Nagata ring \(R[X]_N\) has Noetherian spectrum.
2.
An integral domain \(D\) has locally Noetherian spectrum if and only if the Nagata ring \(D[X]_N\) has locally Noetherian spectrum.
3.
An integral domain \(D\) has \(t\)-locally Noetherian spectrum if and only if the polynomial ring \(D[X]\) has \(t\)-locally Noetherian spectrum, if and only if the \(t\)-Nagata ring \(D[X]_{N_v}\) has \((t\)-\()\)locally Noetherian spectrum.

MSC:

13A15 Ideals and multiplicative ideal theory in commutative rings
13B25 Polynomials over commutative rings
13B30 Rings of fractions and localization for commutative rings
13E99 Chain conditions, finiteness conditions in commutative ring theory
13G05 Integral domains

Citations:

Zbl 0123.03402
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References:

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