Let $G$ be a group, $R$ $G$-graded commutative ring with identity, $M$ graded $R$-module and $S\subseteq h(R)$ multiplicatively closed subset of $R$. In this paper, new concept $S$-primary submodules is introduced as generalization Primary well $S$-prime $M$. Also, some properties class are investigated.

Abstract One of the methods of increasing soil resistance against failure is soil reinforcement using geosynthetics. Soil-geosynthetic interactions are of great importance and are affected by friction and adhesion at their interface. Soil gradation, contact surface roughness and geotextile density are among the factors affecting soil-geotextiles interaction this study, to investigate the eff...

by virtue of a complete set of two displacement potentials, an analytical derivation of the elastostatic green’s functions of an exponentially graded transversely isotropic bi-material full-space was presented. three-dimensional point-load green’s functions for stresses and displacements were given in line-integral representations. the formulation included a complete set of transformed stress-p...

Let $G$ be a group with identity $e$. $R$ $G$-graded commutative ring and $M$ graded $R$-module. In this paper, we introduce the concept of primary-like submodules as new generalization primary ideals give some basic results about modules. Special attention has been paid, when satisfies gr-primeful property, to find extra properties these submodules.

Let G be a group with identity e. R G-graded commutative ring and M graded R-module. We introduce the concept of Ie-prime submodule as generalization prime for I =?g?G Ig fixed ideal R. give number results concerning this class submodules their homogeneous components. A proper N is said to if whenever rg ? h(R) mh h(M) rgmh IeN, then either (N :R M) or N.

We study transcendency properties for graded field extension and give an application to valued field extensions. 1. Introduction. An important tool to study rings with valuation is the so-called associated graded ring construction: to a valuation ring R, we can associate a ring gr(R) graded by the valuation group. This ring is often easier to study, and one tries to lift properties back from gr...

We study the graded derivation-based noncommutative differential geometry of the Z2-graded algebra M(n|m) of complex (n+m)× (n+m)-matrices with the “usual block matrix grading” (for n 6= m). Beside the (infinite-dimensional) algebra of graded forms the graded Cartan calculus, graded symplectic structure, graded vector bundles, graded connections and curvature are introduced and investigated. In...

<abstract><p>Let $ R be a G graded commutative ring and M $-graded $-module. The set of all second submodules is denoted by Spec_G^s(M), it called the spectrum $. We discuss rings with Noetherian prime spectrum. In addition, we introduce notion Zariski socle explore their properties. also investigate Spec^s_G(M) topology from viewpoint being space.</p></abstract>

Let %‘A be the category of finite dimensional right modules for a quasi-hereditary algebra A. In the context of various types of Kazhdan-Lusztig theories, we study both the homological dual A! = Extt, (A/ rad(A), A/ rad(A)) and the graded algebra grA = @rad(A)j/rad(A)j+‘. For example, we investigate a condition introduced in [6], and here called (SKL’), which guarantees that A!’ E gr A. A stren...