Geodesic
On the Symmetric Hypercenter of a Ring
Canadian journal of mathematics, Tome 36 (1984) no. 3, pp. 421-435

Voir la notice de l'article provenant de la source Cambridge University Press

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 = F 2 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. 
Giambruno, A. On the Symmetric Hypercenter of a Ring. Canadian journal of mathematics, Tome 36 (1984) no. 3, pp. 421-435. doi: 10.4153/CJM-1984-026-2
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