Let G be a group with identity e, R be a commutative G-graded ring with unity 1 and M be a G-graded R-module. In this article, we introduce and study two generalizations of graded second submodules, namely, graded 2-absorbing second submodules and graded strongly 2- absorbing second submodules. Also, we introduce and study the concept of graded quasi 2-absorbing second submodules, that is a generalization for graded strongly 2-absorbing second submodules.
Explicit formulae are given for the deflections and stresses in a thin isotropic homogeneous semi‐circular plate subject to an isolated load on the radius of symmetry, clamped along the diameter and free along the circular boundary. Two methods are used and numerical values of deflections, moments and shears based on the theory are provided in tables and curves. Results obtained by the two methods are in good agreement.
Let $G$ be a group with identity $e$, $R$ a $G$-graded commutative ring with unity $1$ and $M$ a $G$-graded $R$-module. In this article, we unify the concepts of graded prime ideals and graded primary ideals into a new concept, namely, graded $\delta$-primary ideals. Also, we unify the concepts of graded $2$-absorbing ideals and graded $2$-absorbing primary ideals into a new concept, namely, graded $2$-absorbing $\delta$-primary ideals. A number of results about graded prime, graded primary, graded $2$-absorbing and graded $2$-absorbing primary ideals are extended into these new structures. Finally, we extend the concept of graded $\delta$-primary ideals into graded $\delta$-primary submodules. A number of results about graded prime, graded primary submodules are extended into this new structure.
ABSTRACT. Let R R 9 be a G-graded ring. In this paper we define the "homogeneousequivalence" concept between graded rings We discuss some properties of the G-graded rings and investigate which of these are preserved under homogeneous-equivalence maps. Furthermore, we give some results in graded ring theory and also some applications of this concept to Z-graded rings
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