A note on the algebra \(P(n)_*(P(n))\) for the prime 2 (Q1078849)

Let BP denote the Brown-Peterson spectrum for the prime 2 and let \(P(n)\) be the spectrum obtained from BP by killing the ideal \(I_ n=(2,v_ 1,...,v_{n-1})\subset \pi *(B)\). \(P(n)\) admits exactly two admissible associative but non-commutative products. We determine the structure of the (commutative) algebra \(P(n)*(P(n))\) and show that several results of \textit{U. Würgler} [Topol. and Algebra, proc. Colloq. in Honor of B. Eckmann, Zürich 1977, 269-280 (1978; Zbl 0398.55003); Manuscr. Math. 29, 93-111 (1979; Zbl 0449.55008] for \(p>2\) carry over to the case \(p=2\). Let k(n) denote the (-1)-connected cover of the nth Morava K-theory \(K(n)\). We show that the structure theorem of \textit{W. Lellmann} [Math. Z. 179, 387-399 (1982; Zbl 0466.55006)] for the algebra \(k(n)*(k(n))\) holds also for \(p=2\) and we remark that \(k(n)\) admits exactly two admissible non-commutative products. In a future publication we shall present a computation of \(k(n)*(k(n))\) for the prime 2.