[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-43292-en":3,"doc-seo-43292-105":30,"detail-sidebar-cat-0-en-105":91},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":21,"is_downloadable":21,"audit_status":21,"page_count":22,"language":23,"language_code":24,"site_id":25,"html_lang":24,"table_of_contents":26,"faqs":27,"seo_title":13,"seo_description":14,"update_tm":28,"read_time":29},43292,1374391975076,"Riley","https://ap-avatar.wpscdn.com/avatar/14000253ca4ec9f6853?x-image-process=image/resize,m_fixed,w_180,h_180&k=1783305029341752051",8,"Research & Report","Commutative Semigroup Rings","Commutative Semigroup Rings presents a structured survey of developments in the theory of commutative semigroup rings. It develops foundational results on commutative semigroups and then studies semigroup rings via key distinguished elements such as zero divisors, nilpotent elements, idempotents, and units. Further chapters establish ring-theoretic properties of monoid domains and monoid rings, including Krull, Prüfer, and factorial behaviors, and address dimension theory and isomorphism problems for monoid rings. The preface emphasizes motivations from group rings and the role of semigroup rings as a unifying concept in commutative algebra.","Chicago Lectures in Mathematics  \ncommutativesemigroupringsRobert Gilmer  \n,UIIiverSILy OI UNicago PressChicago and London  \nChicago Lectures in Mathematics  \nCommutativesemigrouprings  \nRobert Gilmer  \nThe University of Chicago PressChicago and London  \nTo the memory of Tom Parker,who kindled my initial interest incommutative semigroup rings.  \nChicago Lectures in Mathematics Series  \nIrving Kaplansky,Editor  \nThe Theory of Sheaves,by Richard G.Swan(1964)Topics in Ring Theory,by 1.N.Herstein (1969)  \nFields and Rings,by Irving Kaplansky(1969;2d ed.1972)Infinite Abelian Group Theory,by Phillip A.Griffith(1970)Topics in Operator Theory,by Richard Beals(1971)  \nLie Algebras and Locally Compact Groups,by Irving Kaplansky(1971)Several Complex Variables,by Raghavan Narasimhan(1971)Torsion-Free Modules,by Eben Matlis(1973)  \nThe Theory of Benoull Shifts,by Paul C.Shields(1973)  \nStable Homotopy and Generalized Homology,by J.F.Adams(1974)Commutative Rings,by Irving Kaplansky(1974)Banach Algebras,by Richard Mosak (1975)Rings with Involution,by I.N.Herstein (1976)  \nTheory of Unitary Group Representation,by George W.Mackey(1976)Infinite-Dimensional Optimization and Convexity,by lvar Ekeland andThomas Turnbull (1983)  \nThe University of Chicago Press,Chicago 60637The University of Chicago Press,Ltd.,London  \n01984 by The University of ChicagoAll rights reserved.Published 1984Printed in the United States of America  \n9392919089888786858412345  \n1980 Mathematics Subject Classification:13,20M14  \nISBN:0-226-29391-2(cloth)0-226-29392-0(paper)  \nLCN:83-51596  \n.10-10  \nCommutative Semigroup Rings  \nCONTENTS  \nPREFACE………………………………………………………………………………………………………………………………………………vii  \nI. COMMUTATIVE SEMIGROUPS……………………………………………………………………………1  \n1. Basic Notions………………………………………2  \n2. Cyclic Semigroups and Numerical Monoids…………………………9  \n3. Ordered Semigroups………………………………………………………………………22  \n4. Congruences…………………………………………………………………………………………………………28  \n5. Noetherian Semigroups…………………………………………………………………………39  \n6. Factorization in Commutative Monoids…………………………………52  \nII. SEMIGROUP RINGS AND THEIR DISTINGUISHED ELEMENTS..63  \n7. Semigroup Rings………64  \n8. Zero Divisors……………………………………………………………………………………………………82  \n9. Nilpotent Elements………………………………………………………………………………97  \n10. Idempotents……………………………………………………………………113  \n11. Units……………………………………………………………………………………………………………………129  \nIII. RING-THEORETIC PROPERTIES OF MONOID DOMAINS…………………………145  \n12. Integral Dependence for Monoid Domains……………………………147  \n13. Monoid Domains as Prüfer Domains……………………………………………162  \n14. Monoid Domains as Factorial Domains……………………………………171  \n15. Monoid Domains as Krul1 Domains………………………………………………190  \n16. The Divisor Class Group of a Krull Monoid Domain.208  \nIV. RING-THEORETIC PROPERTIES OF MONOID RINGS………………………………225  \n17. Monoid Rings as von Neumann Regular Rings…………………226  \n19. Monoid Rings as Arithmetical Rings一 the Case  \nWhere S is not Torsion-Free…………………………………………………253  \n20. Chain Conditions in Monoid Rings……………………………………………271  \nV. DIMENSION THEORY AND THE ISOMORPHISM PROBLEMS……………………287  \n21. Dimension Theory of Monoid Rings…………………………………………288  \n22. R-Automorphisms of R[S]…………………………………………………………………303  \n23. Coefficient Rings in Isomorphic Monoid Rings. I..317  \n24.Coefficient Rings in Isomorphic Monoid Rings. II.337  \n25. Monoids In Isomorphic Monoid Rings一a Survey..351  \nSELECTED BIBLIOGRAPHY……354  \nTOPIC INDEX………………………………………………………………………………………………………………………………362  \nINDEX OF MAIN NOTATION…………………………………………………………………………………………………370  \nINDEX OF SOME MAIN RESULTS………………………………………………………………………………………373  \nPREFACE  \nThis book contains an account of the current state ofknowledge on some topics in the area of commutative semi-group rings.  Much of the material in the book comes fromliterature of the last ten years,and in this regard,a bitof historical information may be in order.  Work on grouprings goes back at least forty years to Higman's paper[76].Original work in the area focused on two related problems:the is","cbCaigcO7P55e05T","https://ap.wps.com/l/cbCaigcO7P55e05T","pdf",39910243,3,1,396,"English","en",105,"# Contents\n## I. Commutative Semigroups\n## II. Semigroup Rings and Their Distinguished Elements\n## III. Ring-Theoretic Properties of Monoid Domains\n## IV. Ring-Theoretic Properties of Monoid Rings\n## V. Dimension Theory and the Isomorphism Problems\n# Selected Bibliography","[{\"question\":\"What main topics are covered in this book on commutative semigroup rings?\",\"answer\":\"The book covers commutative semigroups, semigroup rings and distinguished elements (zero divisors, nilpotents, idempotents, units), ring-theoretic properties of monoid domains and monoid rings, and applications to dimension theory and isomorphism problems.\"},{\"question\":\"How does the preface motivate the study of commutative semigroup rings?\",\"answer\":\"It motivates semigroup rings as a natural framework in commutative algebra and highlights problems that ask when a semigroup ring R[S] has a desired ring-theoretic property.\"},{\"question\":\"What is the role of monoid rings in the book’s later chapters?\",\"answer\":\"Later chapters analyze ring-theoretic properties of monoid rings, including regularity and arithmetical behavior, and then use dimension theory and structural considerations to address isomorphism 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main topics are covered in this book on commutative semigroup rings?","Question",{"text":75,"@type":76},"The book covers commutative semigroups, semigroup rings and distinguished elements (zero divisors, nilpotents, idempotents, units), ring-theoretic properties of monoid domains and monoid rings, and applications to dimension theory and isomorphism problems.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does the preface motivate the study of commutative semigroup rings?",{"text":80,"@type":76},"It motivates semigroup rings as a natural framework in commutative algebra and highlights problems that ask when a semigroup ring R[S] has a desired ring-theoretic property.",{"name":82,"@type":73,"acceptedAnswer":83},"What is the role of monoid rings in the book’s later chapters?",{"text":84,"@type":76},"Later chapters analyze ring-theoretic properties of monoid rings, including regularity and arithmetical behavior, and then use dimension theory and structural considerations to 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