An algebra extension A | B is right depth two in this paper if its tensor-square is A-B-isomorphic to a direct summand of any (not necessarily finite) direct sum of A with itself. For example, normal subgroups of infinite groups, infinitely generated Hopf-Galois extensions and infinite dimensional algebras are depth two in this extended sense. The added generality loses some duality results obt...

The purpose of this paper is to further the study of weakly injective and weakly projective modules as a generalization of injective and projective modules. For a locally q.f.d. module M , there exists a module K ∈ σ[M ] such that K ⊕N is weakly injective in σ[M ], for any N ∈ σ[M ]. Similarly, if M is projective and right perfect in σ[M ], then there exists a module K ∈ σ[M ] such that K ⊕ N i...

Let p be a prime number. This paper introduces the Roquette category Rp of finite p-groups, which is an additive tensor category containing all finite p-groups among its objects. In Rp, every finite p-group P admits a canonical direct summand ∂P , called the edge of P . Moreover P splits uniquely as a direct sum of edges of Roquette p-groups, and the tensor structure of Rp can be described in t...

We present telescoping algorithms which compute optimal sum representations of indefinite nested sums. More precisely, given a rational summand expression in terms of nested sums and products, the algorithm splits the summand into a summable part, which can be summed by telescoping, and into a non-summable part, which is degree-optimal with respect to one of the most nested sums or products. If...

A composition of a positive integer n consists of an ordered sequence of positive integers whose sum is n. We investigate compositions in which the summand 2 is not allowed, and count the total number of such compositions and the number of occurrences of the summand i in all such compositions. Furthermore, we explore patterns in the values for Cj(n, 2̂), the number of compositions of n without 2...

In this paper we provide a simple characterization of the dominance in two classes of continuous triangular norms. In particular, we solve the dominance of (i) ordinal sum t-norms that use the Łukasiewicz t-norm as the only summand operation and (ii) ordinal sum t-norms that use the product t-norm as the only summand operation. In both cases, the dominance relation is characterized by a simple ...

The method of intersection spaces associates rational Poincaré complexes to singular stratified spaces. For a conifold transition, the resulting cohomology theory yields the correct count of all present massless 3-branes in type IIB string theory, while intersection cohomology yields the correct count of massless 2-branes in type IIA theory. For complex projective hypersurfaces with an isolated...

In this paper, an algebra extension A |B is right depth two if its tensor-square is A-B-isomorphic to a direct summand of any (not necessarily finite) direct sum of A with itself. For example, normal subgroups of infinite groups, infinitely generated Hopf-Galois extensions and infinite dimensional algebras are depth two in this extended sense. The added generality loses some duality results obt...

Let k be a field and let G be a finite group. There is a canonical element in the Hochschild cohomology of the Tate cohomology γG ∈ HH3,−1Ĥ∗(G, k) with the following property. Given a graded Ĥ∗(G, k)-module X, the image of γG in Ext 3,−1 Ĥ∗(G,k) (X,X) vanishes if and only if X is isomorphic to a direct summand of Ĥ∗(G,M) for some kG-module M . The description of the realizability obstruction wo...

We introduce the class of “right almost V-rings” which is properly between the classes of right V-rings and right good rings. A ring R is called a right almost V-ring if every simple R-module is almost injective. It is proved that R is a right almost V-ring if and only if for every R-module M, any complement of every simple submodule of M is a direct summand. Moreover, R is a right almost V-rin...