The aim of this work is to show that (oriented) matroid methods can be applied to many discrete geometries, namely those based on modules over integral (ordered) domains. The trick is to emulate the structure of a vector space within the module, thereby allowing matroid methods to be used as if the module were a vector space. Only those submodules which are “closed under existing divisors,” and...

Let R be a ring and C a class of right R-modules closed under finite direct sums. If we suppose that C has a set of representatives, that is, a set V(C) ⊆ C such that every M ∈ C is isomorphic to a unique element [M ] ∈ V(C), then we can view V(C) as a monoid, with the monoid operation [M1] + [M2] = [M1 ⊕M2]. Recent developments in the theory of commutative monoids (e.g., [4], [15]) suggest tha...

The LDLR (low-density lipoprotein receptor) is a modular protein built from several distinct structural units: LA (LDLR type-A), epidermal growth factor-like and beta-propeller modules. The low pH X-ray structure of the LDLR revealed long-range intramolecular contacts between the propeller domain and the central LA repeats of the ligand-binding domain, suggesting that the receptor changes its o...

Let Λ be a tame hereditary algebra over an algebraically closed field, i.e. Λ = kQ with Q a quiver of type Ãn, D̃n, Ẽ6, Ẽ7, or Ẽ8. Two different kinds of partitions of the module category can be obtained by using Auslander-Reiten theory, and on the other hand, Gabriel-Roiter measure approach. We compare these two kinds of partitions and see how the modules are rearranged according to Gabriel-Roi...

Let $R$ be an arbitrary ring with identity and $M$ a right $R$-module with $S=$ End$_R(M)$. The module $M$ is called {it Rickart} if for any $fin S$, $r_M(f)=Se$ for some $e^2=ein S$. We prove that some results of principally projective rings and Baer modules can be extended to Rickart modules for this general settings.

The singularity category of a ring makes only the modules finite projective dimension vanish among modules, so that is expected to characterize homological property infinite dimension. In this paper, such we deal with eventually periodic over left artin ring, and, as our main result, them in terms morphisms category. As applications, first prove that, for class dimensional algebras field, being...

We introduce a concept of a fractional derivatives series and prove that any linear partial differential equation in two independent variables has a fractional derivatives series solution with coefficients from a differentially closed field of zero characteristic. The obtained results are extended from a single equation to D-modules having infinitedimensional space of solutions (i.e., non-holon...

Let K denote an algebraically closed field with characteristic 0, and let q denote a nonzero scalar in K that is not a root of unity. Let Aq denote the unital associative K-algebra defined by generators x, y and relations xy − [3]qx yx + [3]qxyx 2 − yx = 0, yx − [3]qy xy + [3]qyxy 2 − xy = 0, where [3]q = (q 3−q−3)/(q−q−1). We classify up to isomorphism the finite-dimensional irreducible Aq-mod...

Let F be an algebraically closed field of characteristic p > 0. Suppose that SLn−1(F) is naturally embedded into SLn(F) (either in the top left corner or in the bottom right corner). We prove that certain Weyl modules over SLn−1(F) can be embedded into the restriction L(ω)↓SL n−1(F), where L(ω) is a simple SLn(F)-module. This allows us to construct new primitive vectors in L(ω)↓SL n−1(F) from a...

Suppose S is an affine, noetherian scheme, X is a separated, noetherian S-scheme, E is a coherent OX -bimodule and I ⊂ T (E) is a graded ideal. We study the geometry of the functor Γn of flat families of truncated B = T (E)/I-point modules of length n + 1. We then use the results of our study to show that the point modules over B are parameterized by the closed points of P X2(E). When X = P , w...