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**On continuous extension of differentiatins of ring.**
*(Russian)*
Zbl 0581.13019

Let (A,\(\tau)\) be a topological commutative unitary ring, and A[X] the polynomial ring with coefficients in A and with set of variables X (the variables commuting both with each other and with the coefficients). Then on A[X] there exists a topology \(\tau_ 1\), extending \(\tau\), and making A[X] into a topological ring. Moreover, if \({\mathcal D}\) is a continuous differentiation on (A,\(\tau)\), then there exists a continuous differentiation \({\mathcal D}_ 1\) on \((A[X],\tau_ 1)\), which is n extension of \({\mathcal D}\). Similar results are obtained in the case of a topological field (K,\(\tau)\): (1) for the extension K(\(\alpha)\) of K obtained by adjunction of an (over K) separable element \(\alpha\) ; (2) for the extension K(S) of K obtained by the adjunction of a set S of (over K) separable elements, when \(\tau\) is an \({\mathfrak m}\)-topology and \(| S| <{\mathfrak m}\); (3) for the purely transcendental extension \(\tilde K\) of K obtained by a set X of variables, commuting both with each other and with the elements of K, when \(\tau\) is an \({\mathfrak m}\)- topology, \({\mathfrak m}>\aleph_ 0\).

Reviewer: J.Weinstein

### MSC:

13N05 | Modules of differentials |

13B02 | Extension theory of commutative rings |

13J99 | Topological rings and modules |

13B10 | Morphisms of commutative rings |

12J99 | Topological fields |

13F20 | Polynomial rings and ideals; rings of integer-valued polynomials |