Retraction Note to: N. Zamani “On graded distributive modules,”

On Z-graded associative algebras and their N-graded modules

Let A be a Z-graded associative algebra and let ρ be an irreducible N-graded representation of A on W with finite-dimensional homogeneous subspaces. Then it is proved that ρ(Ã) = glJ (W ), where à is the completion of A with respect to a certain topology and glJ (W ) is the subalgebra of EndW , generated by homogeneous endomorphisms. It is also proved that an N-graded vector space W with finite...

On Torsion Free Distributive Modules

Let R be a commutative ring with identity and let M be a torsion free R-module. Several characterizations of distributive modules are investigated. Indeed, among other equivalent conditions, we prove that M is distributive if and only if any primal submodule of M is irreducible, and, if and only if each submodule of M can be represented as an intersection of irreducible isolated components. MSC...

NUMBERS OF Z n - GRADED MODULES

Let S = K[X 1 ,. .. , X n ] be the polynomial ring over a field K. For bounded below Z n-graded S-modules M and N we show that if Tor S p (M, N) = 0, then for 0 ≤ i ≤ p, the dimension of the K-vector space Tor S i (M, N) is at least p i. In particular, we get lower bounds for the total Betti numbers. These results are related to a conjecture of Buchsbaum and Eisenbud.

A Note on Graded

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A Note on Finsler Modules