Ideal Theory of Commutative Rings and Monoids
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Beschreibung
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.
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.
The first book to cover commutative rings and their modules in the setting of rings AND monoids Develops the theory of ideal systems and weak module systems, and the theory of v-ideals and t-ideals Includes in-depth discussions of Prüfer, Dedekind, and Krull monoids
Autor*in
Franz Halter-Koch
Themen in »Ideal Theory of Commutative Rings and Monoids«
Multiplicative Ideal Theory Ideal Systems Module Systems Prüfer Monoids Prüfer Rings Krull Monoids Krull Rings Mori Monoids Mori Rings Lorenzen Monoids Nagata Rings
Stimmen zu »Ideal Theory of Commutative Rings and Monoids«
“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. Dimitrić, 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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