Hostname: page-component-84b7d79bbc-4hvwz Total loading time: 0 Render date: 2024-08-02T11:24:24.175Z Has data issue: false hasContentIssue false
  • English
  • Français

Radical properties defined locally by polynomial identities I

Published online by Cambridge University Press:  09 April 2009

B. J. Gardner
B. J. Gardner
Affiliation:
Dalhousie UniversityHalifax, N.S., Canada
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Bijective correspondences are established between the radical classes R (in a variety W of rings) with the property that a ring A is in R exactly when its finitely generated subrings are all in R, and certain filters of ideals in a free W-ring. It follows that such classes are determined by the polynomial identities satisfied by the finite subsets of their members. Analogous considerations are applied to radical classes R which, for a fixed integer n, have the property that a ring is in R if and only if its subrings generated by at most n elements are in R.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 27 , Issue 3 , May 1979 , pp. 257 - 273
Copyright
Copyright © Australian Mathematical Society 1979

References

Anderson, T. (1974), ‘On the Levitzki radical’, Canad. Math. Bull. 17, 5–10.CrossRefGoogle Scholar
Freidman, P. A. (1958), ‘On the theory of the radical of an associative ring’, Isv. Vyshch. Ucheb. Zaved. Mat. 3 (4), 225–232 (in Russian).Google Scholar
Gabriel, P. (1962), ‘Des catégories abéliennes’, Bull. Soc. Math. France 90, 323–448.CrossRefGoogle Scholar
Gardner, B. J. (1975), ‘Semi-simple radical classes of algebras and attainability of identities’, Pacific J. Math. 61, 401–416.CrossRefGoogle Scholar
Gardner, B. J. (1977), ‘Polynomial identities and radicals’, Compositio Math. 35, 269–279.Google Scholar
Gardner, B. J. and Stewart, P. N. (1975), ‘On semi-simple radical classes’, Bull. Austral. Math. Soc. 13, 349–353.CrossRefGoogle Scholar
Hu, T. K. (1973), ‘Locally equational classes of universal algebras’, Chinese J. Math. 1, 143–165.Google Scholar
Krempa, J. (1972), ‘Logical connections between some open problems concerning nil-rings’, Fund. Math. 76, 121–130.CrossRefGoogle Scholar
Morse, M. and Hedlund, G. A. (1938), ‘Symbolic dynamics’, Amer. J. Math. 60, 815–864.CrossRefGoogle Scholar
Parfenov, V. A. (1971), ‘On a weakly solvable radical of lie algebras’, Siberian Math. J. 12, 123–127.CrossRefGoogle Scholar
Rivlin, T. J. (1974), The Chebyshev polynomials (Wiley-Interscience, New York).Google Scholar
Ryabukhin, Yu. M. (1965), ‘Semi-strictly hereditary radicals in primitive classes of rings’, Issled. po. Obshchei Algebre, Kishinev, 112–122 (in Russian).Google Scholar
Ryabukhin, Yu. M. (1968), ‘On a class of locally nilpotent rings’, Algebra i Logika 7, 100–108 (in Russian).Google Scholar
Stewart, P. N. (1970), ‘Semi-simple radical classes’, Pacific J. Math. 32, 249–254.CrossRefGoogle Scholar
Stewart, P. N. (1972a), ‘Strongly hereditary radical classes’, J. London Math. Soc. (2) 4, 499–509.CrossRefGoogle Scholar
Stewart, P. N. (1972b), ‘On the locally antisimple radical’, Glasgow Math. J. 13, 42–46.CrossRefGoogle Scholar
Tangeman, R. and Kreiling, D. (1972), ‘Lower radicals in nonassociative rings’, J. Austral. Math. Soc. 14, 419–423.CrossRefGoogle Scholar
Wiegandt, R. (1974), Radical and semisimple classes of rings (Queens' Papers in Pure and Applied Mathematics No. 37, Kingston, Ontario).Google Scholar