Abstract
Smyth proved that given an element x in a semiprime Banach algebra B, xB is finite-dimensional if and only if Bx is finite-dimensional. In a recent paper, Yood extended the result to arbitrary semiprime algebras. In this paper we study unital one-sided ideals in semiprime algebras from different viewpoints and prove several results on equalities of dimensions. Finally, we give a structure theorem of semiprime algebras whose zero subalgebras are finite-dimensional. All arguments concerning algebras can be easily modified to the case of rings.
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