Boca Raton: CRC press, 2016. - 444p.The motivation of this second edition is quite simple: As proofs of PI-theorems have become more technical and esoteric, several researchers have become dubious of the theory, impinging on its value in mathematics. This is unfortunate, since a closer investigation of the proofs attests to their wealth of ideas and vitality. So our main goal is to enable the community of researchers and students to absorb the underlying ideas in recent PI-theory and confirm their veracity. Our main purpose in writing the first edition was to make accessible the in tricacies involved in Kemer’s proof of Specht’s conjecture for affine PI-algebras in characteristic 0. The proof being sketchy in places in the original edition, we have undertaken to fill in all the details in the first volume of this revised edition. In the first edition we expressed our gratitude to Amitai Regev, one of the founders of the combinatoric PI-theory. In this revision, again we would like to thank Regev, for discussions resulting in a tighter formulation of Zubrilin’s theory. Earlier, we thanked Leonid Bokut for suggesting this project, and Klaus Peters for his friendly persistence in encouraging us to complete the manuscript, and Charlotte Henderson at AK Peters for her patient help at the editorial stage. Now we would also like to Rob Stern and Sarfraz Khan of Taylor and Francis for their support in the continuation of this project. Mathematically, we are grateful to Lance Small for the more direct proof (and attribution) of the Wehrfritz–Beidar theorem and other suggestions, and also for his encouragement for us to proceed with this revision. Eli Aljadeff provided much help concerning locating and filling gaps in the proof of Kemer’s difficult PI representability theorem, including supplying an early version of his write-up with Belov and Karasik. Uzi Vishne went over the entire draft and provided many improvements. Finally, thanks again to Miriam Beller for her invaluable assistance in technical assistance for this revised edition.
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