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Combinatorial Algebra: Syntax and Semantics

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Abstract

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.

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Notes

  1. 1.

    Note that every ring identity f = g is equivalent to the identity f − g = 0, so it is enough to consider only identities of the form f = 0

  2. 2.

    Recall that an associative ring with the derived operation (x, y) is a Lie ring. The identity ((x, y), z) = 0 then means that the Lie ring is nilpotent of class 2.

  3. 3.

    If the ring contains 1, then this condition is equivalent to invertibility of 1 + x.

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  1. Department of Mathematics, Vanderbilt University, Nashville, TN, USA

    Mark V. Sapir

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Sapir, M.V. (2014). Rings. In: Combinatorial Algebra: Syntax and Semantics. Springer Monographs in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-08031-4_4

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