Generalizations of the theory of quasi-frobenius rings
Norton, Nicholas Charles
arxiv: Mathematics::Commutative Algebra
In this thesis, we consider several generalizations of the theory of Quasi-Frobenius rings, and construct examples of the classes of rings we introduce. In Chapter 1 we establish well known results, although the way in which we use idempotents is apparently new.\ud \ud Chapter 2 is devoted to the study of three generalizations of Quasi-Frobenius rings, namely D-rings, RD-rings (restricted D-rings) and PD-rings (partial D-rings). PD-rings is the largest class of rings we study, and we show that these rings can be considered as a natural generalization of Nakayama's definition of a Quasi-Frobenius ring. D-rings are defined by annihilator conditions, and RD-rings are a generalization of D-rings. We show that RD-rings, hence also D-rings, are semi-perfect, and it follows that they are also PD-rings. We will show that in the self-injective case, these three classes of rings all coincide with a class of rings studied by Osofsky, . We will investigate when the properties described are Morita invariant, and will show that finitely generated modules over D-rings are finite dimensional. Finally, we study group rings over D-rings, RD-rings and PD-rings, and in particular show that if a group ring is a D-ring, then the group is finite and the ring is a D-ring, and further, either the ring is self-injective or the group is Hamiltonian. In Chapter 3 we construct examples of D-rings, RD-rings and PD-rings.\ud \ud Chapter 4 contains results obtained jointly by Dr. C. R. Hajarnavis and the author. Here, we generalize a result of Hajarnavis, ., by considering Noetherian rings each of whose proper homomorphic imaeges are i.p.r.i.-rings. He will obtain a partial structure theory for such rings, and in the prime bounded case show that such rings are Dedekind prime rings.
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