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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