ON CARTAN-BRAUER-HUA THEOREM
Kaoru Motose · Hokkaido Mathematical Journal · 1967
Since Cartan-Brauer-Hua theorem was generalized as in Jacobson [3, Th. 7. 13. 1] and Nagahara and Tominaga [6, Lemma 2], this theorem has been extended to simple rings (cf.Kishimoto [4] and Nagahara, Kishimoto and Tominaga [5]).The purpose of the present paper is to extend [4, Th. 2] and [5, Th. 2] to primitive rings.Throughout our study, we use the following conventions: $U$ will represent a ring with 1, and $B$ a subdirectly irreducible subring1) of $U$ such that the unique minimal ideal $T$ of $B$ is not nilpotent.A primitive ring and a completely primitive ring will mean a right primitive ring and the ring of all the linear transformations in a left vector space over a division ring, respectively.Let $R$ be a ring.If for any finite subset $F$ of $R$ there exists a completely primitive subring of $R$ containing $F,$ $R$ is said to be locally completely primitive.Let $\Lambda$ be an arbitrary non-empty set.By $(R)_{A}$ and $R^{(A)}$ we denote the ring of all row-finite matrices $(x_{ij})(i,j\in\Lambda)$ and the direct sum of $\#\Lambda^{2)}$ -copies of R-left module $R$ ; thus $(R)_{4}$ can be regarded as the ring of all linear trans- formations in $R^{(A)}$ .We shall first prove the following that contains [4, Th. 2].