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Nonnil-Noetherian rings and the SFT property. (English) Zbl 1242.13027

Let \(R\) be a commutative ring with identity. As in [A. Badawi, Commun. Algebra 31, No. 4, 1669–1677 (2003; Zbl 1018.13010)], we say that \(R\) is a nonnil-Noetherian ring if every ideal of \(R\) that is not contained in Nil\((R)\), the nilradical of \(R\), is finitely generated. In this paper, the authors study nonnil-Noetherian rings and formal power series rings over nonnil-Noetherian rings. For example, they show that \(R[[X]]\) is nonnil-Noetherian if and only if \(R[X]\) is nonnil-Noetherian, if and only if \(R\) is Noetherian, and that Nil\((R[[X]]) =\) Nil\((R)[[X]]\) if and only if Nil\((R)\) is an SFT ideal of \(R\). (An ideal \(I\) of \(R\) is an SFT ideal if there is a finitely generated ideal \(J \subseteq I\) of \(R\) and a positive integer \(k\) such that \(x^k \in J\) for every \(x \in I\); and \(R\) is an SFT ring if every ideal of \(R\) is an SFT ideal.) They also show that if \(R\) is a nonnil-Noetherian SFT ring, then dim\(R[[X_1, \ldots, X_n]] =\) dim\(R + n\).

MSC:

13F25 Formal power series rings
13A15 Ideals and multiplicative ideal theory in commutative rings
13C15 Dimension theory, depth, related commutative rings (catenary, etc.)
16N40 Nil and nilpotent radicals, sets, ideals, associative rings

Citations:

Zbl 1018.13010
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References:

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