JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org.Introduction. Let y be a differential indeterminate over the rational number field R, that is, we consider the polynomial ring R [yO, yl, Y2, * ] in a sequence of (algebraic) indeterminates yo =y, Yl, * * * together with the mapping a--a' of R [yO, Yl, Y2, * * * I into itself which has the properties:(1) (a +b)' = a'+b', (2) (ab)' = a'b +ab', (3) yJ = yi+i; there is one and only one such mapping, and the operation of passing from a to a' is called disfferentiation. By a differential ideal in R [yo, Yl, Y2, ] we mean an ideal in the ringtheoretic sense which has the property that if a is in the ideal, then also a' is in the ideal. Notationally, [yP] stands for the differential ideal generated by yP, that is, for the ideal generated in the usual ring-theoretic sense by yP, (yP), ((Yp)'), *
A study of the structure of differential ideals yields many unsolved problems even for the relatively simple ideal [yP]. It is shown from a simple calculation that y2 -O=0[yP], whence it follows that some power of each yi is in [yP]. The following question was singled out for investigation by J. F. Ritt [3 ]: what is the smallest q such that y = O[yP]? For i= 1, q=2p-1 is stated by him without proof to be the answer. In Part I we give a proof of this result, and in Part II we solve the problem for i =2, p > 2. For arbitrary i we conjecture the answer to be q=(i+1)(p-1)+1.The following notation and results of H. Levi we use extensively. Let P o yy' . . . y'n be a power product (pp.) of degree n d = ,oi i=O and weight iDl Write d =a(p-1)+b where a and b are integers such that O< a, O
