A note to general notions of the derivation and its applications
In a previous paper [the authors, ``A note to the continuous derivation of fields'', Acta Math. Acad. Sci. Hung. 31, 75--79 (1978; Zbl 0381.44006)] a representation theorem for continuous derivations of fields is proved, which is useful for applications for example in the operational calculus. The term ``derivation'' is used in the sense of differential algebra [\textit{I. Kaplansky}, An introduction to differential algebra. Paris: Hermann (1957; Zbl 0083.03301)], and it is required that the field of complex numbers is included in the field considered and the fields of constants of the derivations contain the complex numbers. In the present note we comprehend the term ``derivation'' more general. Let \({\mathcal R}\) be a subring of a commutative ring \({\mathcal R}^*\). A map \({\mathcal J}\) of \({\mathcal R}\) into \({\mathcal R}^*\) having the properties (1) \({\mathcal J}(a +b) = {\mathcal J}(a) +{\mathcal J}(b)\) and (2) \({\mathcal J}(ab) = {\mathcal J}(a)b+a{\mathcal J}(b)\) for all \(a,b \in {\mathcal R}\), will be called a ``derivation of \({\mathcal R}\) into \({\mathcal R}^*\). We use a special derivation \({\mathcal D}\)- which in several cases can be identified with the so-called algebraic derivation - and prove Theorem 1: Under some assumption (which are natural for polynomial elements from \({\mathcal R}\) every sequentially continuous derivation \({\mathcal J}\) of \({\mathcal R}\) into \({\mathcal R}^*\) with \({\mathcal K(J)\subseteq K(D)}\) - fields of ``constants'' - fulfills the relation \({\mathcal D}(s){\mathcal J}={\mathcal J}(s){\mathcal D}\). A similar theorem can be proved for algebras. The second part of the paper gives sufficient conditions for the representation \({\mathcal J=}a{\mathcal D}\). Finally we give several examples: the Mikusiński operators, the space of differentiable functions (example 3.), the Banach algebra of holomorphic functions for \(|z| <1\) (example 4.). We should remark that the derivation type depends on the multiplication defined on the ring (or algebra) being considered.
scientific article; zbMATH DE number 3557904