Let p be a prime and suppose that K/F is a cyclic extension of degree p with group G. Let J be the FpG-module K/K of pth-power classes. In our previous paper we established precise conditions for J to contain an indecomposable direct summand of dimension not a power of p. At most one such summand exists, and its dimension must be p +1 for some 0 ≤ i < n. We show that for all primes p and all 0 ...

This theorem, already known for finitely generated projective modules[l, I, Proposition 6.1], has been recently proved for arbitrary projective modules over commutative semi-hereditary rings by I. Kaplansky [2], who raised the problem of extending it to the noncommutative case. We recall two results due to Kaplansky: Any projective module (over an arbitrary ring) is a direct sum of countably ge...

Given a JBW*-triple Z and a normal contractive projection P : Z −→ Z, we show that the (Murray-von Neumann) type of each summand of P (Z) is dominated by the type of Z.

the class of ads modules with the sip (briefly‎, ‎$sa$-modules) is studied‎. ‎various conditions for a module to be $sa$-module are given‎. ‎it is proved that for a quasi-continuous module $m$‎, ‎$m$ is a uc-module if and only if $m$ is an $sa$-module‎. ‎also‎, ‎it is proved that the direct sum of two $sa$-modules as $r$-modules is an $sa$-module when $r$ is the sum of the annihilators of these...

a p–local CW–complex with H∗X a finite dimensional Fp–vector space. Then there is an integer NX (see 2.5) depending on H ∗X as a graded vector space and: (1) For p = 2, if P s t H ∗X 6= 0 for all P s t ∈ A with s < t, then X∧NX has a non–trivial stable summand Y such that H∗Y is A–free. (2) For p 6= 2, if P s t HX 6= 0 for all P s t ∈ A with s < t and QtHX 6= 0 for all Qt ∈ A, then X∧NX has a n...

Abstract We extend the classical Euler–Maclaurin (EM) expansion to sums over multidimensional lattices that involve functions with algebraic singularities. This offers a tool for precise and fast evaluation of singular appear in long-range interacting systems. find approximation error decays exponentially order band-limited runtime is independent number particles. First, EM summation formula ge...

We provide an argument that (6) of the main paper is an unbiased estiamting function for ξ when the coarsening probabilities may not be correctly specified, so that the posited probabilities π{r, Gr(Z)} 6 = π0{r, Gr(Z)} for some r = 1, . . . ,M,∞, but the models hr{Gz(Z), ξ} are correct, and that estimators for ξ based on (6) will converge in probability to ξ0 for arbitrary choice of the functi...