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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 $$A$$.