Generalizations of the theory of quasi-frobenius rings

Doctoral thesis English OPEN
Norton, Nicholas Charles
  • Subject: QA
    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, [30]. 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, [14]., 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.
  • References (20)
    20 references, page 1 of 2

    Osaka Nath. J. 1 (1949), pp.52 - 61. ASANO, K.: 'Zur Arithmetik in ┬ĚSchiefringen, II',

    I (1950), pp. 1 - 27. BASS, H.: 'Finitistic Dimension and a Homological

    Hath. Soc. 95 (1960), pp. 466 - 488. BJOHK, J. -E.: 'langs satisfying certain chain

    conditions', J. fur Reine und Ange\1. l-fath.

    245 (1970), pp. 63 - 73. CONHELL, 1. G.:'On the group rine', Canad. J. Haths.

    15 (1963), pp. 650 - 685. CURTIS, C. l/. and RBIlIEn, 1.: 'Hepresentution 'l'heory of

    Interscience Publications, 1962. DESH:e1UWE, N. G.:' Structure of riCht subdirectly

    irreducible rings. I.', J. Algebra 17 (1971)~

    pp. 317 - 325. DHAN,. H.: Thesis, Uni v. of Leeds, 1967. [11] GOLDIE, A. H.: 'Rings .\'lith Haximum Condition',

    Multigraphed notes, Yale University, 1961. [12J GOLDIE, A. \'i.:'Non-commutative Principal Ideal Rings',

  • Bioentities (1)
    1lrj Protein Data Bank
  • Metrics
    0
    views in OpenAIRE
    0
    views in local repository
    14
    downloads in local repository

    The information is available from the following content providers:

    From Number Of Views Number Of Downloads
    Warwick Research Archives Portal Repository - IRUS-UK 0 14
Share - Bookmark