Simple graded rings, nonassociative crossed products and Cayley–Dickson doublings

We show that if a nonassociative unital ring is graded by a hypercentral group, then the ring is simple if and only if it is graded simple and the center of the ring is a field. Thereby, we extend a result by Jespers to a nonassociative setting. By applying this result to nonassociative crossed products, we obtain nonassociative analogues of results by Bell, Jordan and Voskoglou. We also apply our result to Cayley–Dickson doublings, thereby obtaining a new proof of a classical result by McCrimmon.

On twisted group -algebras associated with FC-hypercentral groups and other related groups

We show that the twisted group $C^{\ast }$-algebra associated with a discrete FC-hypercentral group is simple (respectively, has a unique tracial state) if and only if Kleppner’s condition is satisfied. This generalizes a result of Packer for countable nilpotent groups. We also consider a larger class of groups, for which we can show that the corresponding reduced twisted group $C^{\ast }$-algebras have a unique tracial state if and only if Kleppner’s condition holds.

Simple semigroup graded rings

We show that if R is a, not necessarily unital, ring graded by a semigroup G equipped with an idempotent e such that G is cancellative at e, the nonzero elements of eGe form a hypercentral group and Re has a nonzero idempotent f, then R is simple if and only if it is graded simple and the center of the corner subring f ReGe f is a field. This is a generalization of a result of Jespers' on the simplicity of a unital ring graded by a hypercentral group. We apply our result to partial skew group rings and obtain necessary and sufficient conditions for the simplicity of a, not necessarily unital, partial skew group ring by a hypercentral group. Thereby, we generalize a very recent result of Gonçalves'. We also point out how Jespers' result immediately implies a generalization of a simplicity result, recently obtained by Baraviera, Cortes and Soares, for crossed products by twisted partial actions.

GROUPS THAT INVOLVE FINITELY MANY PRIMES AND HAVE ALL SUBGROUPS SUBNORMAL II

It is shown that if G is a hypercentral group with all subgroups subnormal, and if the torsion subgroup of G is a π-group for some finite set π of primes, then G is nilpotent. In the case where G is not hypercentral there is a section of G that is much like one of the well-known Heineken-Mohamed groups. It is also shown that if G is a residually nilpotent group with all subgroups subnormal whose torsion subgroup satisfies the above condition then G is nilpotent.

ARTINIAN MODULES OVER FC-HYPERCENTRAL GROUPS

Artinian modules over a group ring DG, where G is an FC-hypercentral group and D is a Dedekind domain, are studied. In particular, the conditions under which an artinian DG-module is countably generated over D, and when the socular height of A is at most, have been obtained.

HYPERCENTRAL GROUPS WITH ALL SUBGROUPS SUBNORMAL III

It is shown that a hypercentral group that has all subgroups subnormal and every non-nilpotent subgroup of bounded defect is nilpotent. As a consequence, a hypercentral group of length at most ω in which every subgroup is subnormal is nilpotent.

A splitting theorem for abelian-by-FC-hypercentral groups

We extend splitting theorems due to Zaicev and Duan proving the following result. Let G be a locally soluble FC-hypercentral group and let A be a periodic artinian ℤG-module. If A has no finite ℤG-submodules then any extension E of A by G splits conjugately over A.1991 Mathematics Subject Classification 20F19.