We generalize this theorem when the motive M(X) ∈ CM(F ; Λ) is replaced by a direct summand (M(X), p) associated with a projector p ∈ EndCM(F ;Λ)(M(X)). The proof given by N. Karpenko in [2] cannot be used in the case where M(X) is replaced by a direct summand because of the use on the multiplicity ([1], §75) as the multiplicity of a projector in the category CM(F ; Λ) is not always equal to 1 ...

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...

Let R be a commutative Noetherian local ring with residue class field k. In this paper, we mainly investigate direct summands of the syzygy modules of k. We prove that R is regular if and only if some syzygy module of k has a semidualizing summand. After that, we consider whether R is Gorenstein if and only if some syzygy module of k has a G-projective summand.

Let k be a field and let Q be a (finite) quiver. We fix a representation S with EndkQ S = k and Ext 1 kQ(S, S) = 0. In analogy to [3, Section 1] we consider the following subcategories of repk Q. Let M S be the full subcategory of all modules X with ExtkQ(S,X) = 0 such that, in addition, X has no direct summand which can be embedded into some direct sum of copies of S. Similarly, let MS be the ...

We describe the main ideas of Ray Heitmann’s proof of the Direct Summand Conjecture in dimension 3 for a ring of mixed characteristic [1]. In the first section we describe the main methods which are used and prove the necessary lemmas. In the second section we prove the main result of Heitmann’s paper. Finally, in the third section we give a proof of the Canonical Element Conjecture using this ...

Let R be ring and M a right R-module. This article introduces the concept of τ −⊕-supplemented modules as follows: Given a hereditary torsion theory in Mod-R with associated torsion functor τ we say that a module M is τ −⊕-supplemented when for every submodule N of M there exists a direct summand K of M such that M = N +K and N ∩K is τ−torsion, and M is called completely τ −⊕-supplemented if ev...

We study Farrell Nil-groups associated to a finite order automorphism of a ring R. We show that any such Farrell Nil-group is either trivial, or infinitely generated (as an abelian group). Building on this first result, we then show that any finite group that occurs in such a Farrell Nil-group occurs with infinite multiplicity. If the original finite group is a direct summand, then the countabl...