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Rings with Central Idempotent or Nilpotent Elements

Published online by Cambridge University Press:  20 January 2009

M. P. Drazin
M. P. Drazin
Affiliation:
Department of Mathematics, North Western University, Evanston, Illinois, U.S.A.
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It is easy to see (cf. Theorem 1 below) that the centrality of all the nilpotent elements of a given associative ring implies the centrality of every idempotent element; and (Theorem 7) these two properties are in fact equivalent in any regular ring. We establish in this note various conditions, some necessary and some sufficient, for the centrality of nilpotent or idempotent elements in the wider class of π-regular rings (in Theorems 1, 2, 3 and 4 the rings in question are not even required to be π-regular).

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 9 , Issue 4 , May 1958 , pp. 157 - 165
Copyright
Copyright © Edinburgh Mathematical Society 1958

References

REFERENCES

[1] Arens, R. and Kaplansky, I., “Topological representation of algebras”, Trans. American Math. Soc., 63 (1948), 457–481.CrossRefGoogle Scholar
[2] Artin, E., Nesbitt, C. J. and Thrall, K. M., Rings with Minimum Condition (University of Michigan, 1944).Google Scholar
[3] Baer, R., “Inverses and zero-divisors”, Bull. American Math. Soc, 48 (1942), 630–638.Google Scholar
[4] Divinsky, N., “Pseudo-regularity”, Canadian J. Math., 7 (1955), 401–410.Google Scholar
[5] Drazin, M. P., “Engel rings and a result of Herstein and Kaplansky”, American J. Math., 77 (1955), 895–913.Google Scholar
[6] Drazin, M. P., “Algebraic and diagonable rings”, Canadian J. Math., 8 (1956), 341–354.CrossRefGoogle Scholar
[7] Forsythe, A. and McCoy, N. H., “On the commutativity of rings”, Bull. American Math. Soc., 52 (1946), 523–526.Google Scholar
[8] Herstein, I. N., “The structure of a certain class of rings”, American J. Math., 75 (1953), 864–871.Google Scholar
[9] Kaplansky, I., “A theorem on division rings”, Canadian J. Math., 3 (1951), 290–292.Google Scholar
[10] Kaplansky, I., “Topological representation of algebras II”, Trans. American Math. Soc., 68 (1950), 62–75.Google Scholar
[11] Kaplansky, I., “Semi-simple alternative rings”, Portugaliae Mathematicae, 10 (1951), 37–50.Google Scholar
[12] Kovàcs, L., “A note on regular rings”, Publ. Math. Debrecen, 4 (1956), 465–468.Google Scholar
[13] Levitzki, J., “On the structure of algebraic algebras and related rings”, Trans. American Math Soc., 74 (1953), 384–409.Google Scholar
[14] McCoy, N. H., “Generalized regular rings”, Bull. American Math. Soc., 45 (1939), 175–178.CrossRefGoogle Scholar
[15] McLaughlin, J. E. and Rosenberg, A., “Zero-divisors and commutativity of rings”, Proc American Math. Soc., 4 (1953), 203–212.Google Scholar