Ideal Theory of Commutative Rings and Monoids
Description
This book offers a concise treatment of multiplicative ideal theory in the language of multiplicative monoids. It presents a systematic development of the theory of weak ideal systems and weak module systems on arbitrary commutative monoids. Examples of monoids that are investigated include, but are not limited to, Mori monoids, Laskerian monoids, Prüfer monoids and Krull monoids. An in-depth study of various constructions from ring theory is also provided, with an emphasis on polynomial rings, Kronecker function rings and Nagata rings. The target audience is graduate students and researchers in ring and semigroup theory.
Reviews / Votes
"A number of topics appear in this book for the first time. ... The bibliography list contains 142 items, including earlier works by the author. There are rudimentary subject and symbol indexes. ... This posthumously published book is a welcome addition to an existing literature of ideal theory of commutative rings and monoids." (Radoslav M. Dimitric, zbMATH 1572.13001, 2026)
"The book under review is most noteworthy for describing multiplicative ideal theory in various situations by means of a single term (i.e., weak module systems). That is, it enables the reader to understand the commonalities of three different areas, say, rings, extensions of rings, and monoids from a shared perspective. The book under review is certainly very helpful for all graduate students and researchers who are interested in the multiplicative ideal theory of rings and monoids." (Gyu Whan Chang, Semigroup Forum, Vol. 111 (3), 2025)
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Persons
Franz Halter-Koch was professor emeritus at the University of Graz, Graz, Austria. He is the author of Ideal Systems (Marcel Dekker,1998), Quadratic Irrationals (CRC, 2013), An Invitation to Algebraic Numbers and Algebraic Functions (CRC Press, 2020), Class Field Theory and L-Functions (CRC 2022), and co-author of Non-Unique Factorizations (CRC 2006). He passed away at the end of 2023, just before finalizing this monograph.
Alfred Geroldinger is professor at the University of Graz, Graz, Austria. He has published more than 100 research papers in commutative algebra and additive combinatorics. He is co-author of Non-Unique Factorizations (CRC 2006) and of Combinatorial Number Theory and Additive Group Theory (Birkhäuser 2009).
Andreas Reinhart is a researcher at the University of Graz, Graz, Austria. He has published about 25 research papers in commutative algebra and algebraic number theory.
Content
- 1. Basic Monoid Theory.- 2. The Formalism of Module and Ideal Systems.- 3. Prime and Primary Ideals and Noetherian Conditions.- 4. Invertibility, Cancellation and Integrality.- 5. Arithmetic of Cancellative Mori Monoids.- 6. Ideal Theory of Polynomial Rings.
