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Some generalizations of nilpotence in ring theory Biggs, Richard Gregory


The study of certain series of groups has greatly aided the development and understanding of group theory. Normal series and central series are particularly important. This paper attempts to define analogous concepts in the theory of rings and to study what interrelationships exist between them. Baer and Freidman have already studied chain ideals, the ring theory equivalent of accessible subgroups. Also, Kegel has studied weakly nilpotent rings, the ring theory equivalent of groups possessing upper central series. Some of the more important results of these authors are given in the first three sections of this paper. Power nilpotent rings, the ring theory equivalent of groups possessing lower central series, are defined in section 4. The class of power nilpotent rings is not homomorphically closed. However, it does possess many of the other properties that the class of weakly nilpotent rings has. In section 5 meta* ideal and U*-ring are defined in terms of descending chains of subrings of the given ring. Not every power nilpotent ring is a U*-ring. This is contrary to the result for semigroups. It is also shown that an intersection of meta* ideals is always a meta* ideal. It follows that not every meta* ideal is a meta ideal since the intersection of meta ideals is not always a meta ideal. Section 6 is concerned with rings in which only certain kinds of multiplicative decomposition take place. The rings studied here are called prime products rings and it is proved that all weakly nilpotent and power nilpotent rings are prime products rings. A result given in the section on U-rings suggests that all U-rings may be prime products rings. The class of prime products rings is very large but does not include any rings with a non-zero idempotent. The last section studies ring types which are defined analogously to group types. The study of which ring types actually occur is nearly completed here. Finally, it is shown that every weakly nilpotent ring has a ring type similar to that of some ring which is power nilpotent. This suggests (but does not prove) the conjecture that all weakly nilpotent rings are power nilpotent.

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