Let $R$ be a right artinian ring or a perfect commutative‎‎ring‎. ‎Let $M$ be a noncosingular self-generator $sum$-lifting‎‎module‎. ‎Then $M$ has a direct decomposition $M=oplus_{iin I} M_i$‎,‎where each $M_i$ is noetherian quasi-projective and each‎‎endomorphism ring $End(M_i)$ is local‎.

We construct certain modified highest weight modules which are called quasi in this paper. Using the modules, we introduce a new category of over an affine Lie superalgebra contains projective covers. also prove that both these covers and satisfy vanishing property BRST cohomology.

A t-intersecting constant dimension subspace code C is a set of k-dimensional subspaces in projective space \(\mathrm {PG}(n,q)\), where distinct intersect exactly t-dimensional subspace. classical example such the sunflower, all pass through same t-space. The sunflower bound states that if \(|C| > \left( \frac{q^{k + 1} - q^{t 1}}{q \right) ^2 1\). In this article we will look at case \(t=0\) ...

let $r$ be a right artinian ring or a perfect commutative‎‎ring‎. ‎let $m$ be a noncosingular self-generator $sum$-lifting‎‎module‎. ‎then $m$ has a direct decomposition $m=oplus_{iin i} m_i$‎,‎where each $m_i$ is noetherian quasi-projective and each‎‎endomorphism ring $end(m_i)$ is local‎.

In this paper we show that if Λ = ∐ i≥0 Λ i is a Koszul algebra with Λ 0 isomorphic to a product of copies of a field, then the minimal projective resolution of Λ 0 as a right Λ-module provides all the information necessary to construct both a minimal projective resolution of Λ 0 as a left Λ-module and a minimal projective resolution of Λ as a right module over the enveloping algebra of Λ. The ...

let $mathcal{x}$ be a class of $r$-modules‎. ‎in this paper‎, ‎we investigate ;$mathcal{x}$-injective (projective) and dg-$mathcal{x}$-injective (projective) complexes which are generalizations of injective (projective) and dg-injecti‎‎ve (projective) complexes‎. ‎we prove that some known results can be extended to the class of ;$mathcal{x}$-injective (projective) and dg-$mathcal{x}$-injective ...

This paper is an introduction to dynamics of dianalytic self-maps of nonorientable Klein surfaces. The main theorem asserts that dianalytic dynamics on Klein surfaces can be canonically reduced to dynamics of some classes of analytic self-maps on their orientable double covers. A complete list of those maps is given in the case where the respective Klein surfaces are the real projective plane, ...