Sums of squares in nonreal commutative rings
Since 50 years there appeared a lot of papers on sums of squares in a (commutative) ring \(R\) and on the invariants \(s(R)\) (level) and \(p(R)\) (Pythagoras number). We should perhaps mention the important papers by \textit{M. Peters} [Math. Ann. 195, 309--314 (1972; Zbl 0217.04201); J. Reine Angew. Math. 268/269, 318--323 (1974; Zbl 0287.10034)], \textit{R. Baeza} [Math. Ann. 207, 121--131 (1974; Zbl 0248.13031); Math. Ann. 215, 13--21 (1975; Zbl 0287.13006); Quadratic forms over semilocal rings. Springer, Cham (1978; Zbl 0382.10014); Arch. Math. 33, 226--231 (1979; Zbl 0421.10015)], \textit{Z. D. Dai} et al. [Bull. Am. Math. Soc., New Ser. 3, 845--848 (1980; Zbl 0435.10016)] and \textit{D. W. Hoffmann} [J. Am. Math. Soc. 12, No. 3, 839--848 (1999; Zbl 0921.11018)] in the list of references. In the present paper the authors introduce two new invariants of \(R\) which they call ``metalevel'' and ``hyperlevel''. They allow a more systematic treatment and slightly sharper inequalities between the (four) invariants of \(R\). Several of the earlier proofs can now be simplified. Nevertheless the paper also contains a lot of new examples where the ``exact values'' of the invariants could not be determined.
scientific article; zbMATH DE number 6808420
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