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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.
Inhaltsverzeichnis
- 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.
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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