Hostname: page-component-848d4c4894-5nwft Total loading time: 0 Render date: 2024-06-09T02:38:02.585Z Has data issue: false hasContentIssue false
  • English
  • Français

On the Symmetric Hypercenter of a Ring

Published online by Cambridge University Press:  20 November 2018

A. Giambruno
A. Giambruno*
Affiliation:
Università di Palermo, Palermo, Italy
Rights & Permissions [Opens in a new window]

Extract

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.

The hypercenter theorem [6] asserts that in a ring with no non-zero nil ideals an element commuting with a suitable power of each element of the ring must be central. In this paper we shall be concerned with a similar problem in the setting of rings with involution. Let R be a ring with involution *, let Z denote the center of R and let S = {x ∈ R|x = x*} be the set of symmetric elements in R. We define the symmetric hypercenter of R to be

What can one hope to say about H? That H need not equal Z is clear. For instance, in the ring R = F2 of 2 X 2 matrices over a field, if * is the symplectic involution, all symmetric elements are central, hence H = R but Z ≠ R. Furthermore if R is a noncommutative ring in which every symmetric element is nilpotent then even in this case H = R and Z ≠ R follows.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 36 , Issue 3 , 01 June 1984 , pp. 421 - 435
Copyright
Copyright © Canadian Mathematical Society 1984

References

1. Chacron, M., A generalization of a theorem of Kaplansky and rings with involution, Mich. Math J. 20 (1973), 45–53.Google Scholar
2. Chacron, M., A commutativity theorem for rings with involution, Can. J. Math. 30 (1978), 1121–1143.Google Scholar
3. Chacron, M., Unitaries in simple artinian rings, Can. J. Math. 31 (1979), 542–557.Google Scholar
4. Chacron, M. and Herstein, I. N., Powers of skew and symmetric elements in division rings, Houston J. Math. 1 (1975), 15–27.Google Scholar
5. Felzenszwalb, B. and Giambruno, A., Centralizers and multilinear polynomials in noncommutative rings, J. London Math. Soc. 19 (1979), 417–428.Google Scholar
6. Herstein, I. N., On the hypercenter of a ring, J. Algebra 36 (1975), 151–157.Google Scholar
7. Herstein, I. N., Rings with involution (U. of Chicago Press, Chicago, 1976).Google Scholar
8. Herstein, I. N., A commutativity theorem, J. Algebra 38 (1976), 112–118.Google Scholar
9. Jacobson, N., Structure of rings, Amer. Math. Soc. Coll. Publ. 37 (1976).Google Scholar
10. Misso, P., Commutativity conditions on rings with involution, Can. J. Math. 34 (1982), 17–22.Google Scholar
11. Smith, M., Rings with an integral element whose centralizer satisfies a polynomial identity, Duke Math. J. 42 (1975), 137–149.Google Scholar