At the prime two, the dual of the Steenrod algebra is a polynomial algebra in generators c,n n > 1. The Eilenberg-Mac Lane spectrum K(Z2) has homology Z2[, In > 1], the Brown-Peterson spectrum BP has homology Z2g2 In > 1], and the symplectic Thom spectrum MSp has homology Z2g4 In > 1] ? E; . In this paper, we show that there is no spectrum Bk with 2k HBk =Z2[ In > 1] for k > 2. In this paper, all spectra are localized at the prime two, and all coefficients are Z2. All our spectra E have units ,u: S -* E, and all our ring spectra have a homotopy unit, are homotopy associative, and are homotopy commutative. Let %* = Z2 [1 in I 1] denote the dual of the Steenrod algebra. Recall [2, 7] that as algebras and f*-comodules, H* KZ2 = Z2[In In? 1 ], H* BP =Z22 In > 1], and H*MSp = Z2k[4 In > 1]? 6, where 6 = Z2[Vmlm :A 2q -1]. The Vm are 2(*-primitive elements of degree 4m. In this paper we show that for k > 2 there is no ring spectrum Bk such that H*Bk = Z2 [4n In > 1] as algebras and W-*-comodules. For k > 4, we prove the stronger result that there is no spectrum 2k Bk such that H*9fk = Z21[kn In > 110 ek as 2f*-comodules, where ek is a set of 2(*-primitive elements. Of course, MSp is an example of a IZ2. We cannot determine whether any 93 exist. If spectra of the type ~Bk, k > 3, had existed, they would have defined generalized Adams spectral sequences which S would have been efficient methods for computing 7r . (For example, see [6] for a description of the MSp-Adams-Novikov spectral sequence for 7r*.) Assume that Bk exists with k > 2. Consider the Adams spectral sequence: (A) E2 = Ext4,(H Bk X Z2)n => 7r*Bk Note that since Bk may not be a ring spectrum, the Adams spectral sequence (A) may not have a multiplicative structure. However, H*Bk = Z21[Rn In > 1] is Received by the editors March 9, 1990. 1980 Mathematics Subject Classification (1985 Revision). Primary 55P42; Secondary 55N22, 55S10, 55T15. This research was partially supported by a grant from the Natural Sciences and Engineering Research Council of Canada. ? 1991 Amencan Mathematical Society 0002-9939/91 $1.00 + $.25 per page