On semirings whose simple semimodules are projective

We study the semirings whose simple semimodules are all projective. In particular, we establish that for every semiring S this condition implies the injectivity of all simple S-semimodules and show that, in contrast to the case of rings, the projectivity of all simple semimodules in general is not a left-right symmetric property.     

Projective semimodules over semirings with valuations in nonnegative integers

We introduce semirings with valuations in nonnegative integers and prove that all projective semimodules over them are free.     

Direct sums and direct products of canonical pencils of matrices

A theorem of Kulikov characterizes the K[x]-modules which are direct sums of finite-dimensional (as a K-vector space) indecomposable modules, where K[x] is the polynomial ring over the field K. In this paper an analogous characterization is given for modules over the ring R, arising from pairs of linear transformations between a pair of complex vector spaces, (V, W). R is a certain subring of the ring of 3 × 3 complex matrices. The equivalence between the category of right R-modules and the category of systems enables one to work entirely in the category of systems. (A pair of complex vector spaces is a system if and only if there is a $\text{C}$-bilinear map from $\text{C}{}^2\text{× V}$ to W). R-modules that are direct sums of finite-dimensional indecomposable subsystems are called pure-projective. The above characterization of pure-projective R-modules is used to prove that an R-module M is projective if and only if Ext(M, R) = 0. Direct products of finite-dimensional indecomposable R-modules are also studied, and a theorem pinpoints those that are pure-projective. An example of an R-module M that is not pure-projective, but with the property that every finite subset of M is contained in a pure-projective direct summand of M, is given. A by-product of this example is a class of matrices that generalizes the Vandermonde matrices.     

Tensor products of direct sums

A similar formula to the one established by Ansemil and Floret for symmetric tensor products of direct sums is proved for alternating and Jacobian tensor products. It is then applied to stable spaces where a number of isomorphisms between spaces of tensors or multilinear forms are unveiled. A connection between these problems and irreducible group representations is made.     

Extending modules which are direct sums of injective modules and semisimple modules

It is proved that, if R is a right Noetherian ringM ₁ is an injective right R-module and M ₂ is a semisimple right R-module, then the right R-module M ₁ + M ₂ is extending if and only if M ₂ is (M ₁/Soc(M ₁))-injective.     

Extreme preservers of maximal column rank inequalities of matrix sums over semirings

We characterize linear operators that preserve sets of matrix ordered pairs which satisfy extreme properties with respect to maximal column rank inequalities of matrix sums over semirings. This work was supported by the research grant of the Cheju National University in 2007.     

On gorenstein injective and projective comodules

Gorenstein injective, projective and coflat comodules were introduced and studied by Asensio and Enochs et al. We further investigate these comodules and introduce n-Gorenstein injective, projective and coflat comodules, which give a new characterization of Gorenstein comodules.     

Cartan-Eilenberg projective,injective and flat complexes

Let R be an associative ring with identity and F a class of R-modules. In this article: we first give a detailed treatment of Cartan-Eilenberg F complexes and extend the basic properties of the class F to the class CE(F). Secondly, we study and give some equivalent characterizations of Cartan-Eilenberg projective, injective and flat complexes which are similar to projective, injective and flat modules, respectively. As applications, we characterize some classical rings in terms of these complexes, including coherent, Noetherian, von Neumann regular rings, QF rings, semisimple rings, hereditary rings and perfect rings.