Jan Krempa

On the composition of derivations

Posner ([9]) has shown that for any prime ringR of characteristic different from 2 the composition of any two non-zero derivations is not a derivation. On the other hand, it is well known ([4]) that if charR=n for a prime numbern andd is a derivation ofR, thendn is also a derivation.Our main objective is to extend the above mentioned result of Posner in the case of commutative domains, and to apply this results to the investigation of connections either between derivations and a center, or between derivations and a generalized centroid of a prime ring. For this purpose, we are first going to introduce a method of notation for the composition of derivations which, we hope, will also be useful in other situations.

On Algebraic Derivations of Prime Rings

It is well known that if R is a ring of characteristic p>0 and d is a derivation of R, then dp is also a derivation. On the other hand, for a prime ring R, powers, less than char R, of inner derivations which are inner derivations were investigated in [3]. It appeared in particular that elements which determined such derivations have to be algebraic and the power of a derivation is not often a derivation.

Gelfand-Kirillov dimensions of a tensor product

On etudie la dimension de Gelfand-Kirillov du produit tensoriel de deux algebres A et B en fonction des dimensions de Gelfand-Kirillov de A et B

Some Examples of Affine Monoid Algebras

We construct a finitely generated monoid S with a zero element such that for every field K the Jacobson radical of the monoid algebra K[S] is a sum of nilpotent ideals but is not nilpotent. Moreover, the contracted monoid algebra K 0[S] is a monomial algebra. If K is a field of characteristic p > 0, then we construct a finitely presented group H p such that the Jacobson radical J of the group algebra K[H p ] is a sum of nilpotent ideals, but is not nilpotent. Moreover, K[H p ]/J is a domain.

Rings with Periodic Unit Groups

Let A be a torsion free abelian group and let G be the group of its automorphisms. Hallett and Hirsch proved that, if G is finite then G has to be a subdirect product of some copies of 6 explicitly distinguished small groups.

A note on indecomposable modules

In this note we study rings having only a finite number of non isomorphic uniform modules with non zero socle. It is proved that a commutative ring with this property is a direct sum of a finite ring and a ring of finite representation type. In the non commutative case we show that most P.I. rings having only a finite number of non isomorphic modules with non zero socle are in fact artinian.

‘ON SETS OF PP-GENERATORS OF FINITE GROUPS’

The classes of finite groups with minimal sets of generators of fixed cardinalities, named

On Finite Number of Conjugacy Classes in Groups

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ON LATTICES OF RADICALS IN THE CLASS OF ALL FINITE GROUPS

n -groups, and groups with the basis property, in which every subgroup is a

Some Examples of Indecomposable Modules

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