## 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
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