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A class of regularities for rings

Published online by Cambridge University Press:  09 April 2009

C. Roos
C. Roos
Affiliation:
Mathematical DepartmentUniversity of TechnologyDelft, Holland
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Abstract

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In this paper a general concept of regularity for rings is defined. It is shown that every regularity determines in a natural way a subradical and a radical for rings. A wide class of regularities is constructed: the class of polynomial regularities. All well-known regularities, such as the Perlis-Jacobson regularity, the von Neumann regularity and many others, belong to this class. Special attention is paid to regularities which are elementary in the sense that the so-called unic and nullic polynomial regularities can be thought of as intersections of the elementary ones.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 27 , Issue 4 , June 1979 , pp. 437 - 453
Copyright
Copyright © Australian Mathematical Society 1979

References

Arens, R. F. and Kaplansky, I. (1948), ‘Topological representations of algebras’, Trans. Amer. Math. Soc. 63, 457–481.CrossRefGoogle Scholar
Baer, R. (1943), ‘Radical ideals’, Amer. J. Math. 65, 537–568.CrossRefGoogle Scholar
Blair, R. L. (1955), ‘A note on f-regularity in rings’, Proc. Amer. Math. Soc. 6, 511–515.Google Scholar
Brown, B. and McCoy, N. H. (1947), ‘Radicals and subdirect sums’, Amer. J. Math. 69, 46–58.CrossRefGoogle Scholar
Brown, B. and McCoy, N. H. (1948), ‘The radical of a ring’, Duke Math. J. 15, 495–499.CrossRefGoogle Scholar
Brown, B. and McCoy, N. H. (1950), ’, Trans. Amer. Math. Soc. 69, 302–311.CrossRefGoogle Scholar
Brown, B. and McCoy, N. H. (1950a), ‘The maximal regular ideal of a ring’, Proc. Amer. Math. Soc. 1, 165–171.CrossRefGoogle Scholar
Divinsky, N. (1955), ‘Pseudo-regularity’, Canad. J. Math. 7, 401–410.CrossRefGoogle Scholar
Divinsky, N. (1958), ‘D-regularity’, Proc. Amer. Math. Soc. 9, 62–71.Google Scholar
Divinsky, N. (1965), Rings are radicals (University of Toronto Press).Google Scholar
Goulding, T. L. and Ortiz, A. H. (1971), ‘Structure of semiprime (p, q) radicals’, Pacific J. Math. 37, 97–99.CrossRefGoogle Scholar
Jacobson, N. (1945), ‘The radical and semi-simplicity for arbitrary rings’, Amer. J. Math. 67, 300–320.CrossRefGoogle Scholar
Kando, T. (1952), ‘Strong regularity in arbitrary rings’, Nogoya Math. J. 4, 51–53.CrossRefGoogle Scholar
Lajos, S. and Szász, F. (1970), ‘Characterizations of strongly regular rings’, Proc. Japan Acad. 46, 38–40.CrossRefGoogle Scholar
McKnight, J. D. and Musser, G. L. (1972), ‘Special (p; q) radicals’, Canad. J. Math. 24, 38–44.CrossRefGoogle Scholar
Musser, G. L. (1971), ‘Linear semiprime (p; q) radicals’, Pacific J. Math. 37, 749–757.CrossRefGoogle Scholar
von Neumann, J. (1936), ‘On regular rings’, Proc. Nat. Acad. Sci. U.S.A. 22, 707–713.CrossRefGoogle ScholarPubMed
Ortiz, A. H. (1971), ‘An intersection theorem for a class of Brown-McCoy radicals’, Tamkang J. Math. 2, 117–121.Google Scholar
Perlis, S. (1942), ‘A characterization of the radical of an algebra’, Bull. Amer. Math. Soc. 48, 128–132.CrossRefGoogle Scholar
Roos, C. (1974), ‘The radical property of nonassociative rings such that every homomorphic image has no nonzero left annihilating ideals’, Math. Nachr. 64, 385–391.CrossRefGoogle Scholar
Roos, C. (1975), Regularities of rings (Dissertation, University of Technology, Delft).Google Scholar
Roos, C. (1976), ‘Ideals in matrixrings over nonassociative rings’, Acta, Math. Acad. Sci. Hung. Tomus 27 (1–2), 7–20.CrossRefGoogle Scholar
de la Rosa, B. (1970), Ideals and radicals (Dissertation, University of Technology, Delft).Google Scholar
Sogowa, M. (1971), ‘On strongly regular rings’, Proc. Japan Acad. 47, 180.Google Scholar
Szász, F. A. (1971), ‘The radical property of rings such that every homomorphic image has no nonzero left annihilators’, Math. Nachr. 48, 371–375.CrossRefGoogle Scholar
Szász, F. A. (1973), ‘A second almost subidempotent radical for rings’, Coll. Math. Soc. János Bolyai 6. Rings, Modules and Radicals, Keszthely (Hungary), 483–499.Google Scholar
Szász, F. A. (1974), ‘Further characterizations of strongly regular rings’, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 22, 243–245.Google Scholar