COMMUNICATIONS IN ALGEBRA, 4 ( 7 ) , 647-656 (1976) A NOTE ON SEMIPRIMITIVITY OF ORE EXTENSIONS C. R. Jordan and D. A. Jordan School of Mathematics and Department of Pure Mathematics The University The University Leeds Shef f ield LS2 9JT S10 2TN England England 91. A well known result on polynomial rings states that, for a given ring R, if R has no non-zero nil ideals then the poly- nomial ring RCXI is semiprimitive, see for example CS - I p.12. In this note we study Ore extensions of the form RCX,UI, where a is an automorphism on the ring R, with the aim of relating the question of the semiprimitivity of RCX,UI to the presence of non- zero nil ideals in R. In particular we show that under certain chain conditions the Jacobson radical of RCX,UI consists precise- ly of polynomials over the nilpotent radical of R. Without restriction on R we show that if u has finite order then ~Cx,al i s semiprimitive if R has no nil ideals. Some conditions are required on R and a for results of the above nature to be true, as is illustrated in 55 by an example of a semiprimitive ring R having an automorphism a of infinite order such that R[x,a] has nil ideals. It is hoped that the results of this paper will appear in the Ph.D. theses of the authors. Copyright @ 1976 by Marcel Dekker, Inc. All Rights Reserved. Neither this work nor any part may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, microfilming, and recording, or by any information storage and retrieval system, without permission in writing from the publisher.