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Multiplicative Ideal Theory in Noncommutative Rings

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Multiplicative Ideal Theory and Factorization Theory

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 170))

Abstract

The aim of this paper is to survey noncommutative rings from the viewpoint of multiplicative ideal theory. The main classes of rings considered are maximal orders, Krull orders (rings), unique factorization rings, generalized Dedekind prime rings, and hereditary Noetherian prime rings . We report on the description of reflexive ideals in Ore extensions and Rees rings. Further we give necessary and sufficient conditions (or sufficient conditions) for well-known classes of rings to be maximal orders, and we propose a polynomial-type generalization of hereditary Noetherian prime rings.

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Acknowledgments

This work has been supported by TUBITAK (project no: 113F032) and by JSPS (project no: 24540058). We would like to thank TUBITAK and JSPS for their support. We would also like to thank Professor Alfred Geroldinger and his students for their warm hospitality and efforts to organize the conference. Our thanks go to the referee who carefully checked the manuscript and gave us so many valuable comments.

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Authors and Affiliations

  1. Department of Mathematics, Hacettepe University, Beytepe Campus, 06800, Ankara, Turkey

    Evrim Akalan

  2. Faculty of Sciences and Engineering, Tokushima Bunri University, Sanuki, Kagawa, 769-2193, Japan

    Hidetoshi Marubayashi

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  1. Evrim Akalan
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  2. Hidetoshi Marubayashi
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Correspondence to Evrim Akalan .

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Editors and Affiliations

  1. Dept of Math & Stat, Sam Houston State University, Huntsville, Texas, USA

    Scott Chapman

  2. Dipartimento di Matematica e Fisica, Università degli Studi Roma Tre, Rome, Italy

    Marco Fontana

  3. Institut für Mathematik und Wissenschaftliches Rechnen, Universität Graz, Graz, Austria

    Alfred Geroldinger

  4. Dept. Mathematical Sciences, New Mexico State University, Las Cruces, New Mexico, USA

    Bruce Olberding

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Akalan, E., Marubayashi, H. (2016). Multiplicative Ideal Theory in Noncommutative Rings. In: Chapman, S., Fontana, M., Geroldinger, A., Olberding, B. (eds) Multiplicative Ideal Theory and Factorization Theory. Springer Proceedings in Mathematics & Statistics, vol 170. Springer, Cham. https://doi.org/10.1007/978-3-319-38855-7_1

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