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On a Theorem of Herstein

Published online by Cambridge University Press:  20 November 2018

M. Chacron
M. Chacron*
Affiliation:
Queen's University, Kingston, Ontario; University of Windsor, Windsor, Ontario
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Throughout this paper, Z is the ring of integers, ƒ*(t) (ƒ(t)) is an integer monic (co-monic) polynomial in the indeterminate t (i.e., each coefficient of ƒ* (ƒ) is in Z and its highest (lowest) coefficient is 1 (5, p. 121, Definition) and M* (M) is the multiplicative semigroup of all integer monic (co-monic) polynomials ƒ* (ƒ) having no constant term. In (3, Theorem 2), Herstein proved that if R is a division ring with centre C such that

1

then R = C. In this paper we seek a generalization of Herstein's result to semi-simple rings. We also study the following condition:

(1)*

Our results are quite complete for a semi-simple ring R in which there exists a bound for the codegree ofƒ (ƒ*) (i.e., the degree of the lowest monomial of ƒ(ƒ*)) appearing in the left-hand side of (1) ((1)*).

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 21 , 1969 , pp. 1348 - 1353
Copyright
Copyright © Canadian Mathematical Society 1969

References

1. Chacron, M., Certains anneaux périodiques, Bull. Soc. Math. Belgique 20 (1968), 66–77.Google Scholar
2. Chacron, M., On quasi periodic rings, J. Algebra 12 (1969), 49–60.Google Scholar
3. Herstein, I. N., The structure of a certain class of rings, Amer. J. Math. 75 (1953), 866–871.Google Scholar
4. Herstein, I. N., Anoie on rings with central nilpotent elements, Proc. Amer. Math. Soc. 5 (1954), 620.Google Scholar
5. Herstein, I. N., Topics in algebra (Blaisdell, Waltham, Massachusetts, 1964).Google Scholar
6. Herstein, I. N., Topics in ring theory, Mathematics Lecture Notes, University of Chicago, Chicago, Illinois, 1965.Google Scholar
7. Rosenberg, A. and Zelinsky, D., On Nakayama1 s extensions of the xn(-x) theorems, Proc. Amer. Math. Soc. 5 (1954), 484–486.Google Scholar
8. Thierrin, G., Extensions radicales et quasi-radicales dans les anneaux, Can. Math. Bull. 5 (1962), 29–35.Google Scholar
9. Utumi, Y., On brings, Osaka Math. J. 33 (1957), 63–65.Google Scholar