Summary We formalize in the Mizar system [3], [4] some basic properties on left module over a ring such as constructing via of endomorphism an abelian group and set all homomorphisms modules form [1] along with Ch. 2 set. 1 [2]. The formalized items are shown below list notations: M ab for Abelian suffix “ ” without is used ring. 1. denoted by End ( ). 2. A pair homomorphism <m:math xmlns:m="ht...

The aim of the present study was to evaluate the strong or weak aspects of an interactive study module introduced during the "Cardiovascular and Respiratory Systems Subject Committee" in the second year of the medical program. Five study groups consisting of 25 students attended two-hour module sessions for six weeks with the same tutor. According to the module assessment questionnaire, the maj...

in this paper we study the module contractibility ofbanach algebras and characterize them in terms the conceptssplitting and admissibility of short exact sequences. also weinvestigate module contractibility of banach algebras with theconcept of the module diagonal.

Let V be a finite-dimensional F -vector space, and let T : V → V be a linear transformation. In this note we prove that V is a direct sum of cyclic T -invariant subspaces. More specifically, we prove that V is a direct sum of cyclic T -invariant subspaces whose annihilators are generated by powers of irreducible polynomials, and that the collection of these polynomials is uniquely determined. R...

Consider the ring R := Q[τ, τ−1] of Laurent polynomials in the variable τ . The Artin’s Pure Braid Groups (or Generalized Pure Braid Groups) act over R, where the action of every standard generator is the multiplication by τ . In this paper we consider the cohomology of such groups with corefficients in the module R (it is well known that such cohomology is strictly related to the untwisted int...

In this work the definition of codes as modules over skew polynomial rings of automorphism type is generalized to skew polynomial rings, whose multiplication is defined using an automorphism and a derivation. This produces a more general class of codes which, in some cases, produce better distance bounds than module skew codes constructed only with an automorphism. Extending the approach of Gab...

The aim of this work is to study a very special family odd-quadratic Lie superalgebras g = 0 ¯ ⊕ 1 such that weak filiform -module (weak type). We introduce concept after having proved the unique non-zero superalgebra ( , B ) with abelian 2-dimensional dim . Let us note in context role center crucial. Thus, we obtain an inductive description type via generalized odd double extensions. Moreover,...

Let $H$ be a weak Hopf algebra that is finitely generated module over its affine center. We show has finite self-injective dimension and so the Brown--Goodearl Conjecture holds in this special setting.

The category k-BFAlg of all k-algebras with a bilinear form, whose objects are pairs (R,b) where R is k-algebra and b:R×R→k mapping, equivalent to the unital A for which canonical homomorphism (k,1)→(A,1A) splitting monomorphism in k-Mod. Call left inverses this “weak augmentations” algebra. There isomorphism between weak augmentation (A,bA) form bA compatible multiplication A, i.e., such that ...

Let K be a field of characteristic pi^O, to exclude trivial cases) and let G be a finite goup. A XG-module M is a finite dimensional K-vector space, on which G acts Jf-linearly from the left. The Green ring a(G) of G(w.r.t. K) is the free abelian group, spanned by the isomorphism classes of indecomposable i£G-modules, with the multiplication induced from the tensor product © x of KGmodules (see...