On the structure of semiprime rings

Abstract

The structure of prime rings has recently been studied by A. W. Goldie, R. E. Johnson, L. Lesieur and R. Croisot. In their main results some sort of finiteness assumption is invariably made. It is shown in this paper that certain semiprime rings are subdirect sums of full rings of linear transformations of a right vector space over a division ring. No finiteness assumption is made about the ring. An apparently new radical property is defined and some of its properties are established; e.g., the radical of a matrix ring R n {R_n} is the matrix ring of the radical of R R .