Elementary radical classes

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.

Keywords

• Radical class,locally equational,Mal'tsev-Neumann product

References

1. B. Amberg and O. Dickenschied, On the adjoint group of a radical ring, Canad. Math. Bull., 38(3) (1995), 262-270.
2. 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.]
3. E. P. Armendariz and J. W. Fisher, Regular P.I.-rings, Proc. Amer. Math. Soc., 39 (1973), 247-251.
4. H. D. Block and H. P. Thielman, Commutative polynomials, Quart. J. Math., Oxford Ser. (2), 2 (1951), 241-243.
5. M. Chacron, On a theorem of Herstein, Canad. J. Math., 21 (1969), 1348-1353.
6. R. C. Courter, Rings all of whose factor rings are semi-prime, Canad. Math. Bull., 12 (1969), 417-426.
7. M. P. Drazin, Algebraic and diagonable rings, Canad. J. Math., 8 (1956), 341- 354.
8. P. Gabriel, Des categories abeliennes, Bull. Soc. Math. France, 90 (1962), 323- 448.

9. B. J. Gardner, Radical properties dened locally by polynomial identities, I, J. Austral. Math. Soc. Ser. A, 27(3) (1979), 257-273.
10. 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.
11. B. J. Gardner, A note on Mal'tsev-Neumann products of radical classes, Int. Electron. J. Algebra, to appear.
12. B. J. Gardner and P. N. Stewart, On semisimple radical classes, Bull. Austral. Math. Soc., 13(3) (1975), 349-353.
13. 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.
14. 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.
15. T. K. Hu, Locally equational classes of universal algebras, Chinese J. Math., 1(2) (1973), 143-165.
16. N. Jacobson, Structure theory for algebraic algebras of bounded degree, Ann. of Math., 46 (1945), 695-707.
17. E. Jacobsthal,  Uber vertauschbare polynome, Math. Z., 63 (1955), 243-276.
18. 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.
19. A. I. Mal'tsev, Multiplication of classes of algebraic systems, Russian, Sibirsk. Mat. Zh., 8 (1967), 346-365.
20. H. Neumann, Varieties of Groups, Springer-Verlag, New York, Inc., New York, 1967.
21. J. M. Osborn, Varieties of algebras, Advances in Math., 8 (1972), 163-369.
22. 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.
23. 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.
24. P. N. Stewart, Semi-simple radical classes, Pacic J. Math., 32 (1970), 249-254.
25. P. N. Stewart, Strongly hereditary radical classes, J. London Math. Soc. (2), 4 (1972), 499-509.
26. F. Szasz, A class of regular rings, Monatsh. Math., 75 (1971), 168-172.
27. W. J. Wickless, A characterization of the nil radical of a ring, Pacic J. Math., 35 (1970), 255-258.