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Linear equations in nilpotent Lie rings. (Equations linéaires dans les anneaux nilpotents au sens de Lie.) (French) Zbl 0849.17005
Let \(A\) be a ring and \(\{L_n (A) \}_{n\geq 1}\) the family of ideals defined by \(L_1 (A)= A\) and \(L_n (A)= A[ L_{n-1} (A), A]\) where \([x, y]= xy- yx\). Assume that \(\bigcap_{n\geq 1} L_n (A)= \{0\}\) and regard the family \(\{L_n (A) \}_{n\geq 1}\) as a fundamental system of neighborhoods of 0 for a separated topology such that \(A\) is a topological ring and let \(\widehat {A}= \varprojlim (A/ L_n (A))\) be the completion of \(A\). If \(a_1, \dots, a_k, b_1, \dots, b_k\in A\), the author proves that the equation \(a_1 x b_1+ \dots+ a_k x b_k =c\) is solvable in \(\widehat {A}\) for all \(c\in \widehat {A}\) if and only if \(a_1 b_1+ \dots+ a_k b_k\) is invertible in \(A\).
Reviewer: M.Boral (Adana)
MSC:
17B30 Solvable, nilpotent (super)algebras
16W99 Associative rings and algebras with additional structure
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