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

Autorentext

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.

Inhalt

• 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.