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Levels of rings — a survey

Volume 108 / 2016

Detlev W. Hoffmann Banach Center Publications 108 (2016), 105-131 MSC: Primary: 11E25; Secondary 11E39. DOI: 10.4064/bc108-0-9


Let $R$ be a ring with $1\neq 0$. The level $s(R)$ of $R$ is the least integer $n$ such that $-1$ is a sum of $n$ squares in $R$ provided such an integer exists, otherwise one defines the level to be infinite. In this survey, we give an overview on the history and the major results concerning the level of rings and some related questions on sums of squares in rings with finite level. The main focus will be on levels of fields, of simple noncommutative rings, in particular division rings, and of arbitrary commutative rings. We also address several variations of the notion of level that have been studied in the literature.


  • Detlev W. HoffmannFakultät für Mathematik
    Technische Universität Dortmund
    D-44221 Dortmund, Germany

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