Research Article
PDF EndNote BibTex RIS Cite

Year 2018, Volume 23, Issue 23, 25 - 41, 11.01.2018
https://doi.org/10.24330/ieja.373634

Abstract

References

  • B. Amberg and O. Dickenschied, On the adjoint group of a radical ring, Canad. Math. Bull., 38(3) (1995), 262-270.
  • V. A. Andrunakievic, Radicals of associative rings I, Amer. Math. Soc. Transl. (Ser. 2), 52 (1966), 95-128. [Russian original: Mat. Sb. N.S., 44(86) (1958), 179-212.]
  • E. P. Armendariz and J. W. Fisher, Regular P.I.-rings, Proc. Amer. Math. Soc., 39 (1973), 247-251.
  • H. D. Block and H. P. Thielman, Commutative polynomials, Quart. J. Math., Oxford Ser. (2), 2 (1951), 241-243.
  • M. Chacron, On a theorem of Herstein, Canad. J. Math., 21 (1969), 1348-1353.
  • R. C. Courter, Rings all of whose factor rings are semi-prime, Canad. Math. Bull., 12 (1969), 417-426.
  • M. P. Drazin, Algebraic and diagonable rings, Canad. J. Math., 8 (1956), 341- 354.
  • P. Gabriel, Des categories abeliennes, Bull. Soc. Math. France, 90 (1962), 323- 448.
  • B. J. Gardner, Radical properties de ned locally by polynomial identities, I, J. Austral. Math. Soc. Ser. A, 27(3) (1979), 257-273.
  • B. J. Gardner, Radical Theory, Pitman Research Notes in Mathematics Series, 198, Longman Scienti c & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1989.
  • B. J. Gardner, A note on Mal'tsev-Neumann products of radical classes, Int. Electron. J. Algebra, to appear.
  • B. J. Gardner and P. N. Stewart, On semisimple radical classes, Bull. Austral. Math. Soc., 13(3) (1975), 349-353.
  • B. J. Gardner and R. Wiegandt, Radical Theory of Rings, Monographs and Textbooks in Pure and Applied Mathematics, 261, Marcel Dekker, Inc., New York, 2004.
  • I. N. Herstein, Noncommutative Rings, The Carus Mathematical Monographs, 15, Published by The Mathematical Association of America; distributed by John Wiley & Sons, Inc., New York, 1968.
  • T. K. Hu, Locally equational classes of universal algebras, Chinese J. Math., 1(2) (1973), 143-165.
  • N. Jacobson, Structure theory for algebraic algebras of bounded degree, Ann. of Math., 46 (1945), 695-707.
  • E. Jacobsthal,  Uber vertauschbare polynome, Math. Z., 63 (1955), 243-276.
  • A. A. Klein, On Fermat's theorem for matrices and the periodic identities of Mn(GF(q)), Arch. Math. (Basel), 34(5) (1980), 399-402.
  • A. I. Mal'tsev, Multiplication of classes of algebraic systems, Russian, Sibirsk. Mat. Zh., 8 (1967), 346-365.
  • H. Neumann, Varieties of Groups, Springer-Verlag, New York, Inc., New York, 1967.
  • J. M. Osborn, Varieties of algebras, Advances in Math., 8 (1972), 163-369.
  • T. J. Rivlin, Chebyshev Polynomials, From approximation theory to algebra and number theory, Second edition, Pure and Applied Mathematics, John Wiley & Sons, Inc., New York, 1990.
  • Yu. M. Ryabukhin, Semistrictly hereditary radicals in primitive classes of rings, Russian, Studies in General Algebra (Sem.) (Russian), Akad. Nauk Moldav. SSR, Kishinev, (1965), 111-122.
  • P. N. Stewart, Semi-simple radical classes, Paci c J. Math., 32 (1970), 249-254.
  • P. N. Stewart, Strongly hereditary radical classes, J. London Math. Soc. (2), 4 (1972), 499-509.
  • F. Szasz, A class of regular rings, Monatsh. Math., 75 (1971), 168-172.
  • W. J. Wickless, A characterization of the nil radical of a ring, Paci c J. Math., 35 (1970), 255-258.

Elementary radical classes

Year 2018, Volume 23, Issue 23, 25 - 41, 11.01.2018
https://doi.org/10.24330/ieja.373634

Abstract

A radical class R of rings is elementary if it contains precisely
those rings whose singly generated subrings are in R. Many examples of ele-
mentary radical classes are presented, and all those which are either contained
in the Jacobson radical class or disjoint from it are described. Attention is
given to those elementary radical classes which are de nable by composition
subsemigroups of the free ring on one generator. Whether every elementary
radical class is of this form remains an open question.

References

  • B. Amberg and O. Dickenschied, On the adjoint group of a radical ring, Canad. Math. Bull., 38(3) (1995), 262-270.
  • V. A. Andrunakievic, Radicals of associative rings I, Amer. Math. Soc. Transl. (Ser. 2), 52 (1966), 95-128. [Russian original: Mat. Sb. N.S., 44(86) (1958), 179-212.]
  • E. P. Armendariz and J. W. Fisher, Regular P.I.-rings, Proc. Amer. Math. Soc., 39 (1973), 247-251.
  • H. D. Block and H. P. Thielman, Commutative polynomials, Quart. J. Math., Oxford Ser. (2), 2 (1951), 241-243.
  • M. Chacron, On a theorem of Herstein, Canad. J. Math., 21 (1969), 1348-1353.
  • R. C. Courter, Rings all of whose factor rings are semi-prime, Canad. Math. Bull., 12 (1969), 417-426.
  • M. P. Drazin, Algebraic and diagonable rings, Canad. J. Math., 8 (1956), 341- 354.
  • P. Gabriel, Des categories abeliennes, Bull. Soc. Math. France, 90 (1962), 323- 448.
  • B. J. Gardner, Radical properties de ned locally by polynomial identities, I, J. Austral. Math. Soc. Ser. A, 27(3) (1979), 257-273.
  • B. J. Gardner, Radical Theory, Pitman Research Notes in Mathematics Series, 198, Longman Scienti c & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1989.
  • B. J. Gardner, A note on Mal'tsev-Neumann products of radical classes, Int. Electron. J. Algebra, to appear.
  • B. J. Gardner and P. N. Stewart, On semisimple radical classes, Bull. Austral. Math. Soc., 13(3) (1975), 349-353.
  • B. J. Gardner and R. Wiegandt, Radical Theory of Rings, Monographs and Textbooks in Pure and Applied Mathematics, 261, Marcel Dekker, Inc., New York, 2004.
  • I. N. Herstein, Noncommutative Rings, The Carus Mathematical Monographs, 15, Published by The Mathematical Association of America; distributed by John Wiley & Sons, Inc., New York, 1968.
  • T. K. Hu, Locally equational classes of universal algebras, Chinese J. Math., 1(2) (1973), 143-165.
  • N. Jacobson, Structure theory for algebraic algebras of bounded degree, Ann. of Math., 46 (1945), 695-707.
  • E. Jacobsthal,  Uber vertauschbare polynome, Math. Z., 63 (1955), 243-276.
  • A. A. Klein, On Fermat's theorem for matrices and the periodic identities of Mn(GF(q)), Arch. Math. (Basel), 34(5) (1980), 399-402.
  • A. I. Mal'tsev, Multiplication of classes of algebraic systems, Russian, Sibirsk. Mat. Zh., 8 (1967), 346-365.
  • H. Neumann, Varieties of Groups, Springer-Verlag, New York, Inc., New York, 1967.
  • J. M. Osborn, Varieties of algebras, Advances in Math., 8 (1972), 163-369.
  • T. J. Rivlin, Chebyshev Polynomials, From approximation theory to algebra and number theory, Second edition, Pure and Applied Mathematics, John Wiley & Sons, Inc., New York, 1990.
  • Yu. M. Ryabukhin, Semistrictly hereditary radicals in primitive classes of rings, Russian, Studies in General Algebra (Sem.) (Russian), Akad. Nauk Moldav. SSR, Kishinev, (1965), 111-122.
  • P. N. Stewart, Semi-simple radical classes, Paci c J. Math., 32 (1970), 249-254.
  • P. N. Stewart, Strongly hereditary radical classes, J. London Math. Soc. (2), 4 (1972), 499-509.
  • F. Szasz, A class of regular rings, Monatsh. Math., 75 (1971), 168-172.
  • W. J. Wickless, A characterization of the nil radical of a ring, Paci c J. Math., 35 (1970), 255-258.

Details

Journal Section Articles
Authors

B. J. GARDNER This is me

Publication Date January 11, 2018
Published in Issue Year 2018, Volume 23, Issue 23

Cite

Bibtex @research article { ieja373634, journal = {International Electronic Journal of Algebra}, issn = {1306-6048}, eissn = {1306-6048}, address = {1710 Sokak, No:41, Batikent/Ankara}, publisher = {Abdullah HARMANCI}, year = {2018}, volume = {23}, number = {23}, pages = {25 - 41}, doi = {10.24330/ieja.373634}, title = {Elementary radical classes}, key = {cite}, author = {Gardner, B. J.} }
APA Gardner, B. J. (2018). Elementary radical classes . International Electronic Journal of Algebra , 23 (23) , 25-41 . DOI: 10.24330/ieja.373634
MLA Gardner, B. J. "Elementary radical classes" . International Electronic Journal of Algebra 23 (2018 ): 25-41 <https://dergipark.org.tr/en/pub/ieja/issue/33727/373634>
Chicago Gardner, B. J. "Elementary radical classes". International Electronic Journal of Algebra 23 (2018 ): 25-41
RIS TY - JOUR T1 - Elementary radical classes AU - B. J.Gardner Y1 - 2018 PY - 2018 N1 - doi: 10.24330/ieja.373634 DO - 10.24330/ieja.373634 T2 - International Electronic Journal of Algebra JF - Journal JO - JOR SP - 25 EP - 41 VL - 23 IS - 23 SN - 1306-6048-1306-6048 M3 - doi: 10.24330/ieja.373634 UR - https://doi.org/10.24330/ieja.373634 Y2 - 2022 ER -
EndNote %0 International Electronic Journal of Algebra Elementary radical classes %A B. J. Gardner %T Elementary radical classes %D 2018 %J International Electronic Journal of Algebra %P 1306-6048-1306-6048 %V 23 %N 23 %R doi: 10.24330/ieja.373634 %U 10.24330/ieja.373634
ISNAD Gardner, B. J. . "Elementary radical classes". International Electronic Journal of Algebra 23 / 23 (January 2018): 25-41 . https://doi.org/10.24330/ieja.373634
AMA Gardner B. J. Elementary radical classes. IEJA. 2018; 23(23): 25-41.
Vancouver Gardner B. J. Elementary radical classes. International Electronic Journal of Algebra. 2018; 23(23): 25-41.
IEEE B. J. Gardner , "Elementary radical classes", International Electronic Journal of Algebra, vol. 23, no. 23, pp. 25-41, Jan. 2018, doi:10.24330/ieja.373634

Cited By