Sums of squares in nonreal commutative rings

Detlev W. Hoffmann, David B. Leep

Research output: Contribution to journalArticlepeer-review


We investigate elements in a commutative ring that can be written as a sum of squares and we study invariants that measure how many squares are needed in such a representation. We focus on rings where −1 is a sum of squares. For such a ring R, we define the metalevel sm(R) and the hyperlevel sh(R), and we relate these to the classical level s(R) and the Pythagoras number p(R) of the ring. Among many results, we prove that sm(R) ≤ s(R) ≤ p(R) ≤ sh(R) + 1 ≤ sm(R) + 2. We also study generic rings that realize prescribed values for the various levels.

Original languageEnglish
Pages (from-to)803-835
Number of pages33
JournalIsrael Journal of Mathematics
Issue number2
StatePublished - Sep 1 2017

Bibliographical note

Publisher Copyright:
© 2017, Hebrew University of Jerusalem.

ASJC Scopus subject areas

  • General Mathematics


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