JOURNAL OF ALGEBRA 90, 217-219 (1984) An Example of a Primitive Polynomial Ring T. J. HODGES Department of Mathematics, University of Utah, Salt Lake City, Utah 84112 Communicated by P. M. Cohn Received December 15, 1982 In [5, p. 2011 Resco poses the following question: If the polynomial ring R [X] is primitive, must R be primitive ? We answer the question in the negative by exhibiting local rings (i.e., R/Jac R is a division ring) with arbitrarily large finite Krull dimensions (in the sense of Gabriel and Rentschler, see [2]) such that R (X, ,..., X,] is primitive for all positive integers s. Let D be a division ring with center C. Recall the definition of the transcendence degree of D given in [3 1: tr deg,D = sup{tr deg,F: F a subfield of D}. Amitsur and Small show that if A,, is the nth Weyl algebra, its quotient division ring D, has transcendence degree n [ 11. Since D,-, C D, we may take the union of these D, and U, D, is a division ring of infinite transcen- dence degree. Hence there exist division rings of all possible transcendence degrees. Now let T = D[ Y, ,..., Y,] and let P = (Y, ,..., Y,,), the maximal ideal generated by Y, ,..., Y,,. Let S = r\P. By [6,2.2], S is an Ore set. Define R = TS-’ =D[Y,,..., Y,]c, ,,..., y,). Then R has Jacobson radical PR, and R/PR z D so that R is a local ring. By [2,9.4], K dim R < K dim T = n. On the other hand R contains chains of primes of length n, so K dim R must equal n [2,7.9]. THEOREM. Let D be a division ring with center C and let d be a positive integer such that tr deg,D > d. Let R = D[ Y ,,..., Ynlcy ,,..., y,, where n <d. Then for any positive integer s < d - n + 1, R [X, ,..., X,] is primitive. This result follows immediately using a result of Resco’s [4,3.2] from the following proposition. Recall that for a Noetherian domain, the extended center is the center of the quotient division ring. For the rings considered here this will just be the quotient field of the center of the ring. 217 0021.8693/84 $3.00 Copyright Q 1984 by Academic Press, Inc. All rights of reproduction in any form reserved.