Irving Kaplansky's Contributions To Algebra

As a famous American mathematician,Irving Kaplansky was president of the American Mathematical Society,director of the Mathematical Sciences Research Institute,member of the National Academy of Sciences and member of the American Academy of Arts and Sciences.His research field mainly focused on algebra and had achieved fruitful results in the fields of ring theory,module theory and so on.Based on the study of a large number of related original literature and research literature,this dissertation discusses his algebra achievements by using the methods of conceptual analysis,chronicle and literature analysis,taking Kaplansky's representative work on Kurosch's problem,PI ring theory,orthogonal module lattice,infinite Abelian group and Hopf algebra as examples.At the same time,Kaplansky's successful experience in education,scientific research and management are analyzed,so that people can have a clearer understanding of his achievements in algebra and the origin and development of algebra.The main results and conclusions are as follows:1.The background,methods and influence of Kaplansky's solution to the Kurosch's problem are explored.He studied the problem in different ways and gave an affirmative answer,one of which was that he transformed the problem into a nil algebra: a nil algebra satisfying polynomial identities is locally finite.2.Kaplansky's achievements in PI ring theory are introduced.He proved that a primitive algebra with polynomial identity is finite dimensional over its center,which created an important branch of non-commutative algebra.3.Kaplansky's pioneering work in the field of orthomodular lattice,and the origin,development and application of orthomodular lattice are excavated.His most important work in this field was to introduce the concept of “orthogonality” into the field of modular lattices,and for the first time named the orthogonal complement lattice with orthogonal pair as modular pair as “orthomodular lattice”,and proved that any orthocomplemented complete modular lattice is a continuous geometry.4.This dissertation studied Kaplansky's classic “Infinite Abelian Groups”.It is considered that this work contains abundant structure theorems,and its publication changed the research focus from finite Abelian groups to infinite Abelian groups,which promoted the study of infinite Abelian groups.5.The ten conjectures in Hopf algebra put forward by Kaplansky in 1975 and the current development of these conjectures are expounded.To some extent,these conjectures promoted the research and development of Hopf algebra.6.The characteristics of Kaplansky's education and scientific research are analyzed.The results show that he had an open mind,adhered to the inquiry teaching method,encouraged students to read widely and read more masterpieces.He also suggested that researchers should make pioneering mathematical problems with leading quality as guarantee,and at the same time had critical thinking and persistent scientific spirit.7.Kaplansky's achievements during his tenure as president of the American Mathematical Society and director of the Mathematical Sciences Research Institute are examined,as well as his close contacts with Chinese mathematicians such as Shiing-Shen Chern,Loo-keng Hua and Shaoxue Liu.