Hiroaki Komatsu

Generalized Derivations with Invertible Values

Abstract Bergen et al. [Bergen, J., Herstein, I. N., Lanski, C. (1983). Derivations with invertible values. Canad. J. Math. 35:300–310] determined the structure of a ring with invertible derivations. We generalize their results to invertible generalized derivations.

Generalized Derivations of Associative Algebras

In their paper [6], Leger and Luks introduced the notion of a generalized derivation in nonassociative algebras and got several results for generalized derivations of Lie algebras. This generalized derivation is closely related to Breˇsars and the second authors generalized derivations in associative rings. In this note, we discuss the relations of these generalized derivations.

QUASI-SEPARABLE EXTENSIONS OF NONCOMMUTATIVE RINGS

We show that the module of differentials of a separable ring extension vanishes. The result is applied to the theory of skew polynomials rings.

On the module of differentials of a noncommutative algebra and symmetric biderivations of a semiprime algebra

We construct the module of differentials of an algebra extension with respect to left derivations, right derivations and central derivations, and, using it, we characterize symmetric biderivations of 2-torsion free semiprinie rings. In [I], BreSar, Martindale and Miers showed that every biderivation of a noncommutative prime ring R is of the form X[x, y], where X is an element in the extended centroid of R. In particular, it follows that a noncommutative prime ring does not have a nonzero symmetric biderivation. On the other hand, a commutative ring may have a lot of symmetric biderivations. For instance, if di ( a = 1, 2, . . . , n) are derivations of a commutative ring R, then A(x, y) = C;=, d,(x)ct, (y) is a symmetric biderivation of R. The first author of this paper and several authors studied the

HIGH ORDER KÄHLER MODULES OF NONCOMMUTATIVE RING EXTENSIONS

We construct the high order Kähler modules of noncommutative ring extensions B/A and show their fundamental properties. Our Kähler modules represent not only high order left derivations for one-sided modules but also high order central derivations for bimodules, which are usual derivations. This new viewpoint enables us to prove new results which were not known even though B is an algebra over a commutative ring A. Our results are the decomposition of Kähler modules by an idempotent element, exact sequences of Kähler modules, the Kähler modules of factor rings, and the relation to separable extensions. In particular, our exact sequences of high order Kähler modules were not known even though B is commutative.

The Module of Differentials of a Noncommutative Ring Extension

We study elementary properties of modules of differentials of noncommutative ring extensions and give a commutativity condition for separable algebras.