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 ...

In this study, the modules whose p-submodules have a complement which is direct summand are explored. The module theoretical properties such as sums and summands investigated. As opposed to sums, condition does not transfer summands. Thus, it examined that under what conditions fulfill property. Examples given demonstrate results.

A module is said to be $PI$-extending provided that every projection invariant submodule is essential in a direct summand of the module. In this paper, we focus on direct summands and indecomposable decompositions of $PI$-extending modules. To this end, we provide several counter examples including the tangent bundles of complex spheres of dimensions bigger than or equal to 5 and certain hyper ...

There are essential arithmetical differences between algebras which satisfy the i?-condition (i?-algebras) and those which do not, especially with regard to class-number properties (Eichler [l, 2, 3]). The meaning of the i?-condition in the case n = 2 is as follows. Both k and A are simple algebras over the field ko of rational numbers, of orders m and 4m, respectively, over &oSuppose ko is ext...

We give three proofs that valuation rings are derived splinters: a geometric proof using absolute integral closure, homological which reduces the problem to checking splinters (which is done in second author’s PhD thesis and we reprise here), by approximation Bhatt’s of direct summand conjecture. The property also shows smooth algebras over splinters.

In the present paper, we give a short proof of the nuclearity property of a class of CuntzPimsner algebras associated with a HilbertA-bimodule M, whereA is a separable and nuclear C*-algebra. We assume that the left A-action on the bimodule M is given in terms of compact module operators and that M is direct summand of the standard Hilbert module over A.

Let B be a p-block of a finite group with abelian defect group D. Suppose that D has no elementary abelian direct summand of order p. Then we show that B satisfies Brauer’s k(B)-Conjecture (i. e. k(B) ≤ |D|). Together with former results, it follows that Brauer’s k(B)-Conjecture holds for all blocks of defect at most 3. We also obtain some related results.

Suppose a connected reductive complex algebraic group G acts linearly on a complex projective variety X. We prove that if 1→ N → G→ H → 1 is a short exact sequence of connected reductive groups and X the open set of semistable points for the action of N on X then IH H (X/N) is a direct summand of IH G (X). The inclusion is provided by the decomposition theorem and certain resolutions of the act...

Let R be a commutative Noetherian local ring. This paper deals with the problem asking whether R is Gorenstein if the nth syzygy module of the residue class field of R has a non-trivial direct summand of finite Gdimension for some n. It is proved that if n is at most two then it is true, and moreover, the structure of the ring R is determined essentially uniquely.