商品描述
This book offers an exploration of the rich interplay between numerical semigroups and commutative algebra. It fills a notable gap in the literature by bridging numerical semigroup theory with advanced algebraic methods. The book is structured to support both self-study and advanced coursework, and it is divided into two major parts. The first three chapters lay the algebraic groundwork for later applications to numerical semigroups. They introduce readers to foundational topics in commutative algebra, homological methods, Cohen-Macaulay and canonical modules--all with a focus on the graded structures that arise naturally in semigroup rings. Building on the first three chapters, Chapters 4-6 lead to deep results connecting semigroup properties and invariants to algebraic properties and homological data of semigroup rings. Throughout, the exposition is enriched with illustrative examples, detailed proofs, and exercises to reinforce understanding. The book is designed for graduate students in mathematics as well as researchers in algebra, number theory, and combinatorics.
商品描述(中文翻譯)
本書探討了數值半群與交換代數之間的豐富互動。它填補了文獻中的一個顯著空白，通過將數值半群理論與先進的代數方法相結合。該書的結構旨在支持自學和進階課程，並分為兩個主要部分。前三章為後續應用於數值半群奠定了代數基礎。它們向讀者介紹了交換代數、同調方法、Cohen-Macaulay 模塊和典範模塊等基礎主題，所有內容都專注於在半群環中自然產生的分級結構。在前三章的基礎上，第四至第六章引入了深刻的結果，將半群的性質和不變量與半群環的代數性質和同調數據聯繫起來。
整體而言，該書的闡述充滿了示例、詳細的證明和練習，以加強理解。本書適合數學研究生以及代數、數論和組合數學的研究人員。
作者簡介
Jürgen Herzog was a professor of mathematics at the University of Duisburg-Essen, Germany. He received his doctorate at Louisiana State University in 1969 and completed his habilitation at the University of Regensburg in 1974. Since 1975, he was a professor at the University of Essen, later Duisburg-Essen, where he retired in 2009. His main research area was in the field of commutative algebra. During the course of his scientific career, he published more than 250 articles and made a significant impact in the development of commutative algebra. Jürgen Herzog sadly passed away on 23 April 2024 before this book could be published. He will be greatly missed in the mathematical community. Somayeh Moradi is associate professor of mathematics at Ilam University, Iran. She received her doctorate from Amirkabir University of Technology - Tehran Polytechnic in 2009. She was appointed assistant professor at Ilam University in 2010 and later in 2017 as associate professor. From march 2023 to March 2025, she was the recipient of the Alexander von Humboldt Research Fellowship at the University of Duisburg-Essen. She is also the recipient of the Maryam Mirzakhani Award 2025. Her main research interests lie in commutative algebra and combinatorics. Masoomeh Rahimbeigi is a researcher at the University of Duisburg-Essen, Germany. She received her doctorate from the University of Kurdistan, Iran in 2019. Her main research area is in commutative algebra.
作者簡介(中文翻譯)
尤根·赫爾佐克（Jürgen Herzog）是德國杜伊斯堡-埃森大學的數學教授。他於1969年在路易斯安那州立大學獲得博士學位，並於1974年在雷根斯堡大學完成了資格論文（habilitation）。自1975年起，他在埃森大學（後來的杜伊斯堡-埃森大學）任教，並於2009年退休。他的主要研究領域是交換代數。在他的科學生涯中，他發表了超過250篇文章，對交換代數的發展產生了重大影響。尤根·赫爾佐克於2024年4月23日不幸去世，未能在本書出版前見證其成果。他在數學界將被深切懷念。 索馬耶·莫拉迪（Somayeh Moradi）是伊朗伊拉姆大學的數學副教授。她於2009年在德黑蘭工業大學（Amirkabir University of Technology - Tehran Polytechnic）獲得博士學位。她於2010年被任命為伊拉姆大學的助理教授，並於2017年晉升為副教授。從2023年3月到2025年3月，她獲得了杜伊斯堡-埃森大學的亞歷山大·馮·洪堡研究獎學金。她還是2025年瑪莉亞姆·米爾扎哈尼獎的獲得者。她的主要研究興趣在於交換代數和組合數學。 馬蘇美·拉希姆貝基（Masoomeh Rahimbeigi）是德國杜伊斯堡-埃森大學的研究員。她於2019年在伊朗庫德斯坦大學獲得博士學位。她的主要研究領域是交換代數。
