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Trace properties and the rings \(R({\scriptstyle {\mathrm{X}}})\) and \(R\langle {\scriptstyle {\mathrm{X}}}\rangle \)

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Abstract

An integral domain R is an RTP domain (or has the radical trace property) (resp. an LTP domain), if I(R : I) is a radical ideal for each nonzero noninvertible ideal I (resp. \(I(R:I)R_P=PR_P\) for each minimal prime P of I(R : I)). Clearly each RTP domain is an LTP domain, but whether the two are equivalent is open except in certain special cases. In this paper, we study when the Nagata ring \(R({\scriptstyle {\mathrm{X}}})\) and the ring \(R\langle {\scriptstyle {\mathrm{X}}}\rangle \) are LTP (resp. RTP) domains in different contexts of integral domains such as integrally closed domains, Noetherian and Mori domains, pseudo-valuation domains and more. We also study the descent of these notions from particular overrings of R to R itself.

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Acknowledgements

Funding was provided by King Fahd University of Petroleum and Minerals (Grant No. SB181004).

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Authors and Affiliations

  1. Department of Mathematics and Statistics, University of North Carolina Charlotte, Charlotte, NC, 28223, USA

    Thomas G. Lucas

  2. Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals, Dhahran, 31261, Saudi Arabia

    Abdeslam Mimouni

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  1. Thomas G. Lucas
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  2. Abdeslam Mimouni
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Correspondence to Abdeslam Mimouni.

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Lucas, T.G., Mimouni, A. Trace properties and the rings \(R({\scriptstyle {\mathrm{X}}})\) and \(R\langle {\scriptstyle {\mathrm{X}}}\rangle \). Annali di Matematica 199, 2087–2104 (2020). https://doi.org/10.1007/s10231-020-00957-8

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