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On the adjoint group of some radical rings

Published online by Cambridge University Press:  18 May 2009

Oliver Dickenschied
Oliver Dickenschied
Affiliation:
Fachbereich Mathematik, der Universität Mainz, D-55099 Mainz, E-mail: dickenschied@mat.mathematik.uni-mainz.de
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A ring R is called radical if it coincides with its Jacobson radical, which means that Rforms a group under the operation a ° b = a + b + ab for all a and b in R. This group is called the adjoint group R° of R. The relation between the adjoint group R° and the additive group R+ of a radical rin R is an interesting topic to study. It has been shown in [1] that the finiteness conditions “minimax”, “finite Prufer rank”, “finite abelian subgroup rank” and “finite torsionfree rank” carry over from the adjoint group to the additive group of a radical ring. The converse is true for the minimax condition, while it fails for all the other above finiteness conditions by an example due to Sysak [6] (see also [2, Theorem 6.1.2]). However, we will show that the converse holds if we restrict to the class of nil rings, i.e. the rings R such that for any a є R there exists an n = n(a) with an = 0.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 39 , Issue 1 , January 1997 , pp. 35 - 41
Copyright
Copyright © Glasgow Mathematical Journal Trust 1997

References

REFERENCES

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