Rings with Periodic Additive Group in which All Subrings are Ideals

Citation

Kruse, Robert Leroy (1964) Rings with Periodic Additive Group in which All Subrings are Ideals. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/BVXD-X137. https://resolver.caltech.edu/CaltechETD:etd-09302002-085204

Abstract

NOTE: Text of symbols not renderable in plain ASCII are indicated by [...]. Abstract is included in .pdf document. A ring in which every subring is a two sided ideal is called a v-ring. This dissertation is a classification of all v-rings with periodic additive group. It is first shown that a ring is a v-ring with periodic additive group if and only if it is the restricted ring direct sum of v-rings whose additive groups are p-groups for different primes p. Such rings are called p-v-rings. It is next shown that a p-v-ring must be nil, or be isomorphic to the ring of rational integers mod p[superscript n] for some n > 1, or be isomorphic to the direct sum of the prime field of p elements and a nil p-v-ring. The classification of nil p-v-rings constitutes the major part of this dissertation. Nil p-v-rings containing elements of unbounded additive order are first characterised. Redei has shown that for any element x of a nil p-v-ring either (I) x[superscript 2] is a natural multiple of x or (II) px[superscript 2] is a natural multiple of x although x[superscript 2] is not a natural multiple of x. Because of this result it is possible to study a nil p-v-ring possessing a bound on the additive orders of its elements by decomposing the ring into an additive group direct sum of cyclic groups. It is shown that aside from elements in the annihilator of the ring, there is a decomposition of the ring with at most two generators of type (I) and three of type (II). The possible defining relations for these nil p-v-rings are enumerated.

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