Simple and semisimple additive categories are studied. We prove, for example, that an artinian additive category is (semi)simple iff it is Morita equivalent to a division ring(oid). Semiprimitive additive categories (that is, those with zero radical) are those which admit a noether full, faithful functor into a category of modules over a division ringoid. Mathematics Subject Classifications (19...

Let S denote the graded polynomial ring C[x1, . . . , xm]. We interpret a chain complex of free S-modules having artinian homology as an S1-equivariant map Cm \ {0} → X, where X is a moduli space of exact sequences. We obtain cohomological obstructions to such maps, most notably a completely new derivation of Herzog-Kühl-type equations for chain complexes. CONTENTS

Let Λ be a left-artinian ring. Generalizing the Loewy length, we propose the layer length associated with a torsion theory, which is a new measure for finitely generated Λ-modules. As an application, we obtain a theorem having as corollaries the main results of [3] and [7].

‎It is shown that‎ ‎if $M$ is an Artinian module over a ring‎ ‎$R$‎, ‎then $M$ has Noetherian dimension $alpha $‎, ‎where $alpha $ is a countable ordinal number‎, ‎if and only if $omega ^{alpha }+2leq it{l}(M)leq omega ^{alpha‎ +1}$, ‎where $ it{l}(M)$ is‎ ‎the length of $M$‎, ‎$i.e.,$ the least ordinal number such that the interval $[0‎, ‎ it{l}(M))$ cannot be embedded in the lattice of all su...

We apply set-theoretic methods to study projective modules and their generalizations over transfinite extensions of simple artinian rings R. prove that if R is small, then the Weak Diamond implies projectivity an arbitrary module can be tested at layer epimorphisms

We introduce the notion of E-depth graded modules over polynomial rings to measure depth certain Ext modules. First, we characterize with (sufficiently) large as those whose partial general initial submodules preserve Hilbert function local cohomology supported at irrelevant maximal ideal, extending a result Herzog and Sbarra on sequentially Cohen-Macaulay Second, describe cone tables sufficien...