let $r$ be a commutative ring. we write $mbox{hom}(mu_a, nu_b)$ for the set of all fuzzy $r$-morphisms from $mu_a$ to $nu_b$, where $mu_a$ and $nu_b$ are two fuzzy $r$-modules. we make$mbox{hom}(mu_a, nu_b)$ into fuzzy $r$-module by redefining a function $alpha:mbox{hom}(mu_a, nu_b)longrightarrow [0,1]$. we study the properties of the functor $mbox{hom}(mu_a,-):frmbox{-mod}rightarrow frmbox{-mo...

Let $R$ be a commutative ring. We write $mbox{Hom}(mu_A, nu_B)$ for the set of all fuzzy $R$-morphisms from $mu_A$ to $nu_B$, where $mu_A$ and $nu_B$ are two fuzzy $R$-modules. We make$mbox{Hom}(mu_A, nu_B)$ into fuzzy $R$-module by redefining a function $alpha:mbox{Hom}(mu_A, nu_B)longrightarrow [0,1]$. We study the properties of the functor $mbox{Hom}(mu_A,-):FRmbox{-Mod}rightarrow FRmbox{-Mo...

We construct a structure of a ring with local units on a co-Frobenius coalgebra. We study a special class of co-Frobenius coalgebras whose objects we call symmetric coalgebras. We prove that any semiperfect coalgebra can be embedded in a symmetric coalgebra. A dual version of Brauer's equivalence theorem is presented, allowing a characterization of symmetric coalgebras by comparing certain func...

Let R be a discrete valuation ring with a unique maximal ideal πR and a quotient field K, and let F = R/πR be the residue class field. Let n ≥ 2 be an integer and {λij | 1 ≤ i, j ≤ n} a set of n integers satisfying λii = 0, λik + λkj ≥ λij , λij + λji > 0 (if i = j) for all 1 ≤ i, j, k ≤ n. Then Λ = (πijR) is a basic semiperfect Noetherian R-subalgebra of the full n× n matrix algebra Mn(K). We ...

We develop the theory of special biserial and string coalgebras and other concepts from the representation theory of quivers. These theoretical tools are then used to describe the finite dimensional comodules and Auslander-Reiten quiver for the coordinate Hopf algebra of quantum SL(2) at a root of unity. We also describe the stable Green ring. Let C = k ζ [SL(2)] denote the quantized coordinate...

We will study the relationship of quite different object in the theory of artin algebras, namely Auslander-regular rings of global dimension two, torsion theories, τ -categories and almost abelian categories. We will apply our results to characterization problems of Auslander-Reiten quivers. 0.1 There exists a bijection between equivalence classes of Krull-Schmidt categories C with additive gen...

Let m and n be positive integers. The algorithmic search for perfect dominating sets of the rectangular grid Gm;n satisfying an initial condition S 0 de ned as an admissible subset of a side Gm;1 of Gm;n such that S \ Gm;1 = S 0 is considered. A binary decision algorithm that generates all such perfect dominating sets is presented, and some related questions and conjectures are posed, leading t...

We will study the relationship of quite different objects in the theory of artin algebras, namely Auslander-regular rings of global dimension two, torsion theories, τ -categories and almost abelian categories. We will apply our results to characterization problems of Auslander-Reiten quivers. 0.1. There exists a bijection between equivalence classes of Krull-Schmidt categories C with additive g...