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Weighted projective lines, introduced by Geigle and Lenzing in 1987, are important objects in representation theory. They have tilting bundles, whose endomorphism algebras are the canonical algebras introduced by Ringel. The aim of this paper is to study their higher dimensional analogs. First, we introduce a certain class of commutative Gorenstein rings

In this paper, we introduce the concept of quasi - copure submodules which is ageneralization of a copure submodules. Weused this concept to define the class of quasi - coregular module, where an R-module M iscalled quasi -coregular module if every submodule of M is quasi-co-pure. Many results about thisconcept are proved.

In this paper, we studied the resolution of Weyl module for characteristic zero in the case of partition (8,7,3) by using mapping Cone which enables us to get the results without depended on the resolution of Weyl module for characteristic free for the same partition.

The article presents the results of studying the influence of such factors as the contact surface of the phases, the degree of grinding of the material, the nature of the flow and the ratio of the two phases called the hydro-module. It has been experimentally proven that at a hydro-module of 1: 4.1: 5, soluble components diffuse to the surface of the material, mixing and dissolving almost completely. By choosing a more acceptable hydro-module, it is possible to obtain a large percentage yield of the extract with the most valuable components, as well as to intensify the extraction process.

The aim of the paper is to prove that if some extended curvatures on a preinfinitesimal module L , considered in the paper, vanish, then the derived preinfinitesimal module L (1) is a Lie pseudoalgebra. Two non-trivial examples are given. The first example is when L 0 is an infinitesimal module and the second one is when L 1 is a preinfinitesimal module.

We study the representation theory of the increasing monoid. Our results provide a fairly comprehensive picture of the representation category: for example, we describe the Grothendieck group (including the effective cone), classify injective objects, establish properties of injective and projective resolutions, construct a derived auto-duality, and so on. Our work is motivated by numerous connections of this theory to other areas, such as representation stability, commutative algebra, simplicial theory, and shuffle algebras.

In this paper, we present a new method for finding identities for hypergeoemtric series, such as the (Gauss) hypergeometric series, the generalized hypergeometric series and the Appell-Lauricella hypergeometric series. Furthermore, using this method, we get identities for the hypergeometric series

Let M be an R-module and T a submodule of M. The notion of small-essential and T-essential submodule are well known. In this paper, we introduce small-essential submodule of M relative to T, which is a generalization of both ones. Several properties and characterizations have been introduced. We invest this concept to consider two socles of modules and study some of their properties.

Let R be an arbitrary ring and T a submodule of an R-module M. A submodule N is said to be Te-small in M, if for each essential submodule X of M, T ⊆ N + X implies that T ⊆ X. In this work we study this mentioned notion which is a generalization of the essential-small submodules as well as the T-small submodule. We use this notion to investigate T-essential radical of module, also to introduce generalized T-essential Hopfian modules.

We introduce the notion of regularity for a relative holonomic ${\\mathcal{D}}$ -module in the sense of Monteiro Fernandes and Sabbah [Internat. Math. Res. Not. (21) (2013), 4961–4984]. We prove that the solution functor from the bounded derived category of regular relative holonomic modules to that of relative constructible complexes is essentially surjective by constructing a right quasi-inverse functor. When restricted to relative ${\\mathcal{D}}$ -modules underlying a regular mixed twistor ${\\mathcal{D}}$ -module, this functor satisfies the left quasi-inverse property.