## 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

### Keywords:

nilradical; nonnil-Noetherian ring; power series ring; SFT ring

Zbl 1018.13010
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### References:

 [1] J.T. Arnold, Krull dimension in power series rings , Trans. Amer. Math. Soc. 177 (1973), 299-304. · Zbl 0262.13007 [2] —, Power series rings over discrete valuation rings , Pacific J. Math. 93 (1981), 31-33. · Zbl 0438.13016 [3] J.T. Arnold and J.W. Brewer, On flat overrings, ideal transforms and generalized transforms of a commutative ring , J. Algebra 18 (1971), 254-263. · Zbl 0218.13019 [4] A. Badawi, On nonnil-Noetherian rings , Comm. Algebra 31 (2003), 1669-1677. · Zbl 1018.13010 [5] A. Badawi and T.G. Lucas, Rings with prime nilradical , in Arithmetical properties of commutative rings and monoids , Lect. Notes Pure Appl. Math. 241 , Chapman & Hall/CRC, · Zbl 1096.13001 [6] J.W. Brewer and W.D. Nichols, Seminormality in power series rings , J. Algebra 82 (1983), 282-284. · Zbl 0512.13013 [7] J. Coykendall, The SFT property does not imply finite dimension for power series rings , J. Algebra 256 (2002), 85-96. · Zbl 1069.13011 [8] D.E. Fields, Zero divisors and nilpotent elements in power series rings , Proc. Amer. Math. Soc. 27 (1971), 427-433. · Zbl 0219.13023 [9] S. Hizem and A. Benhissi, When is $$A+XB[[X]]$$ Noetherian? , C.R. Acad. Sci. Paris 340 (2005), 5-7. · Zbl 1056.13014 [10] I. Kaplansky, Commutative rings , rev. ed, The University of Chicago Press, Chicago, 1974. · Zbl 0296.13001 [11] J. Ohm and R.L. Pendleton, Rings with Noetherian spectrum , Duke Math. J. 35 (1968), 631-639. · Zbl 0172.32201 [12] J. Querré, Cours d’algèbre , Masson, 1976. · Zbl 0341.00001
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