Skew polynomial rings over semisimple rings

The present paper contains a detailed study of the (right) skew polynomial rings over semisimple rings (see definitions below). In a sequel to this paper, we shall consider skew polynomial rings over right orders in right Artinian rings. The results of this paper as well as its sequel were announced in Ref. [8]. Skew polynomial rings over skew fields have been the subject of study for quite some time; therefore, the rings considered here may be of some interest on their own right. 1Ve briefly indicate the contents and layout of this paper. In Section 1, we recall some definitions and explain some notation. In Section 2 we look at the skew polynomial ring Q[x, p] assuming that Q is a self-basic semisimple ring and p induces a cycle on the set of primitive idempotents of Q. The series considered in Lemma 2.3 may be of independent interest. In Section 3, it is shown that the skew polynomial rings over arbitrary semisimple rings are finite direct sums of matrix rings over the type of rings considered in Section 3. Although we have treated skew polynomial rings only, all our results can be adopted to skew power series rings over semisimple rings; this requires minor changes which are omitted. Some special cases of our results are known [I-3, 12, and 131.