For each prime l with l ≡ 7 (mod 8), we define an action of the ring O = Z[ 2(1 + √ −l)] on the unimodular lattice D l+1 using a Paley matrix. We determine the isomorphism class of D l+1 as an O-module. In particular we show that unless l = 7, D l+1 is not a free O-module. We note a consequence for the Leech lattice.

We fix a prime number p and a number field K, and denote by M the maximal abelian p-extension of K unramified outside p. Our aim is to study the Zp-module X = Gal(M/K) and to give a method to effectively compute its structure as a Zp-module. We also give numerical results, for real quadratic fields, cubic fields and quintic fields, together with their interpretations via Cohen-Lenstra heuristics.

Let R be a ring and let I 6= R be an ideal of R. Then I is said to be a completely prime ideal of R if R/I is a domain and is said to be completely semiprime if R/I is a reduced ring. In this paper, we introduce a new class of rings known as completely prime ideal rings. We say that a ring R is a completely prime ideal ring (CPI-ring) if every prime ideal of R is completely prime. We say that a...

In the building of a finite group of Lie type we consider the incidence relations defined by oppositeness of flags. Such a relation gives rise to a homomorphism of permutation modules (in the defining characteristic) whose image is a simple module for the group. The p-rank of the incidence relation is then the dimension of this simple module. We give some general reductions towards the determin...

Let K be an arbitrary field of characteristic p > 0 and D(Pn) the ring of differential operators on a polynomial algebra Pn in n variables. A long anticipated analogue of the inequality of Bernstein is proved for the ring D(Pn). In fact, three different proofs are given of this inequality (two of which are essentially characteristic free): the first one is based on the concept of the filter dim...

Let H be a bialgebra and let A be an associative algebra. The algebra A is said to be an H-module-algebra if there is an H-module structure on A such that the multiplication on A becomes an H-module morphism. For example, if S denotes the Landweber-Novikov algebra [15, 21], then the complex cobordism MU(X) of a topological space X is an S module-algebra. Likewise, the singular mod p cohomology ...

Let K be an arbitrary field of characteristic p > 0 and D(Pn) be the ring of differential operators on a polynomial algebra Pn in n variables. A long anticipated analogue of the inequality of Bernstein is proved for the ring D(Pn). In fact, three different proofs are given of this inequality (two of which are essentially characteristic free): the first one is based on the concept of the filter ...