Introduction. Let. k be a field. The purpose of this article is to introduce, for each field extension K of k, the concept of the /C-radical of ideals in &-algebras. We show that the /f-radical generalizes the usuai radicai of ideals and that it satisfies almost ali the properties of the usuai radicai. Moreover, we introduce /f-Jacobson algebras, that generalize the Jacobson algebras, and give some of their properties. The reason for introducing the /f-radicai of an ideal is that this radicai appears naturally in a generalization of the Hilbert NuUstellensatz to a NuUstellensatz where the zeroes of the ideals in polynomial rings over k are taken in (affine spaces over) K rather than in the algebraic closure of k. We have in a previous article ("Radicais and the Hilbert NuUstellensatz over not necessarily algebraically closed fieids", Preprint, Uppsala Univ. 1985) given such a generalization. There we did, however, give only those properties needed for the proof of the NuUstellensatz and only for the case when K is an algebraic extension of k. Here we give a more complete treatment, valid for any extension K of k. We hope that the theory developed here, and particularly the introduction of iC-Jacobson rings, will make it possible to generalize the algebraic forms of the Hilbert NuUstellensatz to more refined forms where the "rationality" properties of maximal ideals of the &-algebras with respect to K are taken into account.
1980 AMS Subject Classification: 16 A 66
78 § 1. Definition and the main properties of radicals. We shall throughout this work fix a (commutative) field k and denote by K a field extension of k. Moreover, we shall fix a set of elements yx, y2, y 3 ) . . . that are algebraically independent over k and denote by R(m) = = ^ [yit y2> ••• > y»i] t n e polynomial ring in m variables over k. The subset of R(m) consisting of the homogeneous polynomials whose zeroes in the cartesian product K are of the form ( 0 ^ , 0 : 2 , . . . , 0 ^ ^ , 0 ) ™ we denote by PK(m). That is PK(m) = {p € R(m) \p is homogeneous and if p(«i ,a 2 ,... , a w ) = 0 with OLJEK for 1 = 1,2,..., w, then a w = 0} . EXAMPLE 1. If K contains an algebraic closure of k we clearly have that P K (w) = {ay w |0^=aGfe and i = 0,1,2,...} . DEFINITION 2. Let ^4 b e a &-algebra and / an ideal of A. We define the K-radical of / to be the following subset of A : {a E A |for some positive integer m there exists a polynomial m pEPK(m) and elements alta2,:..tam-i A such that p{alfal9... ,am-ìfa)ei} . The /C-radical of I we denote by V 7 . An ideal / such that shall cali K-radical. Vi = / we
EXAMPLE 3. If K contains an algebraic closure of k it follows from Exampie 1 that \ / 7 = [a E A | am £I for some naturai number m} = y/7 . Hence in this case the K-radical is the usuai radicai of /. EXAMPLE 4. Assume that there is a homomorphism
