• Mark V. Sapir
Part of the Springer Monographs in Mathematics book series (SMM)


In this chapter, all rings are assumed to be associative. We start with a study of free associative algebras. This gives us the main syntactic tool to study rings. The first major result proved in this chapter is Shirshov’s height theorem, which is used to prove a theorem of Kaplansky. Then we prove the Dubnov–Ivanov–Nagata–Higman theorem about associative algebras satisfying the identity x n  = 0. Unlike semigroups, associative rings satisfying this identity are nilpotent. But there exist finitely generated non-nilpotent associative nil-algebras, and we describe the classical example of such an algebra constructed by Golod.


Associative Algebra Homogeneous Polynomial Left Ideal Hilbert Series Associative Ring 
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Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Mark V. Sapir
    • 1
  1. 1.Department of MathematicsVanderbilt UniversityNashvilleUSA

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