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      Radicals of 0-regular algebras

      McConnell, N. R.; Stokes, Tim E.
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      DOI
       10.1007/s10474-006-0087-8
      Link
       www.springerlink.com
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      Citation
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      McConnell, N.R. & Stokes, T. E. (2006) Radicals of 0-regular algebras. Acta Mathematica Hungarica, 16(6), 1131-1159.
      Permanent Research Commons link: https://hdl.handle.net/10289/2010
      Abstract
      We consider a generalisation of the Kurosh--Amitsur radical theory for rings (and more generally multi-operator groups) which applies to 0-regular varieties in which all operations preserve 0. We obtain results for subvarieties, quasivarieties and element-wise equationally defined classes. A number of examples of radical and semisimple classes in particular varieties are given, including hoops, loops and similar structures. In the first section, we introduce 0-normal varieties (0-regular varieties in which all operations preserve 0), and show that a key isomorphism theorem holds in a 0-normal variety if it is subtractive, a property more general than congruence permutability. We then define our notion of a radical class in the second section. A number of basic results and characterisations of radical and semisimple classes are then obtained, largely based on the more general categorical framework of L. M\'arki, R. Mlitz and R. Wiegandt as in [13]. We consider the subtractive case separately. In the third section, we obtain results concerning subvarieties and quasivarieties based on the results of the previous section, and also generalise to subtractive varieties some results for multi-operator group radicals defined by simple equational rules. Several examples of radical and semisimple classes are given for a range of fairly natural 0-normal varieties of algebras, most of which are subtractive.
      Date
      2006
      Type
      Journal Article
      Publisher
      Akademiai Kiado Rt
      Rights
      This is an author’s version of an article published in the journal: Acta Mathematica Hungarica. © 2006 Springer. The original publication is available at www.springerlink.com.
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      • Computing and Mathematical Sciences Papers [1452]
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