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Duplicate of an algebra and a theorem of Whitehead. (Dupliquée d’une algèbre et le théorème de Whitehead.) (French) Zbl 0815.17004
The authors give a construction of the duplicate of an algebra using divided powers [following the notation and terminology of N. Roby, Ann. Sci. Éc. Norm. Supér., III. Sér. 80, 213-348 (1963; Zbl 0117.023)]: \(\Gamma^ 2_ K (A)\) is the sub \(K\)-module of homogeneous elements of degree 2 of the \(K\)-module \(A\), where \(K\) is a commutative ring with unit element and \(A\) a commutative \(K\)-algebra and moreover \(\Gamma^ 2_ K\) is equipped with a suitable algebra structure. If 2 has an inverse in \(K\), this duplicate is equivalent to the duplicate of \(A\) obtained after using symmetric powers. However, if \(2\lambda=0\) in \(K\) for each element \(\lambda \in K\), then the equivalence is not always true, as is shown in several examples.
Finally the authors prove a Whitehead theorem for this duplicate. A historical note on duplicate algebras closes the study.
MSC:
17A99 General nonassociative rings
17D99 Other nonassociative rings and algebras
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