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Zero-divisors in completions of non-commutative rings. (English) Zbl 0784.16025
An example is constructed of a Noetherian ring $$R$$ with the following properties: there is a maximal ideal $$M$$ of $$R$$ such that $$R/M$$ is Artinian and the intersection of the powers of $$M$$ is 0; and there is a regular normal element $$x$$ of $$R$$ such that $$x$$ becomes a zero divisor in the completion of $$R$$ at $$M$$. This contrasts with the fact that if $$R$$ is commutative then $$x$$ remains regular in the completion.
##### MSC:
 16W60 Valuations, completions, formal power series and related constructions (associative rings and algebras) 16P40 Noetherian rings and modules (associative rings and algebras) 16S36 Ordinary and skew polynomial rings and semigroup rings
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##### References:
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