scholarly journals Graded ϕ-2-Absorbing and Graded ϕ-2-Absorbing Primary Submodules

Mathematics ◽  
2021 ◽  
Vol 9 (10) ◽  
pp. 1083
Author(s):  
Azzh Saad Alshehry ◽  
Malik Bataineh ◽  
Rashid Abu-Dawwas
Keyword(s):  
Homogeneous Element ◽  
Primary Submodules

The main goal of this article is to explore the concepts of graded ϕ-2-absorbing and graded ϕ-2-absorbing primary submodules as a new generalization of the concepts of graded 2-absorbing and graded 2-absorbing primary submodules. Let ϕ:GS(M)→GS(M)⋃{∅} be a function, where GS(M) denotes the collection of graded R-submodules of M. A proper K∈GS(M) is said to be a graded ϕ-2-absorbing R-submodule of M if whenever x,y are homogeneous elements of R and s is a homogeneous element of M with xys∈K−ϕ(K), then xs∈K or ys∈K or xy∈(K:RM), and we call K a graded ϕ-2-absorbing primary R-submodule of M if whenever x,y are homogeneous elements of R and s is a homogeneous element of M with xys∈K−ϕ(K), then xs or ys is in the graded radical of K or xy∈(K:RM). Several properties of these new forms of graded submodules are investigated.

NS-Primary Submodules

2018 ◽  
Vol 59 (1B) ◽  
Keyword(s):  
Primary Submodules

Divisibility Properties of Graded Domains

1982 ◽  
Vol 34 (1) ◽  
pp. 196-215 ◽  
Author(s):  
D. D. Anderson ◽  
David F. Anderson
Keyword(s):  
Integral Domain ◽  
Homogeneous Element ◽  
Carry Over ◽  
Divisibility Properties ◽  
Krull Domains

Let R = ⊕α∊гRα be an integral domain graded by an arbitrary torsionless grading monoid Γ. In this paper we consider to what extent conditions on the homogeneous elements or ideals of R carry over to all elements or ideals of R. For example, in Section 3 we show that if each pair of nonzero homogeneous elements of R has a GCD, then R is a GCD-domain. This paper originated with the question of when a graded UFD (every homogeneous element is a product of principal primes) is a UFD. If R is Z+ or Z-graded, it is known that a graded UFD is actually a UFD, while in general this is not the case. In Section 3 we consider graded GCD-domains, in Section 4 graded UFD's, in Section 5 graded Krull domains, and in Section 6 graded π-domains.

On $n-$absorbing primary submodules

2021 ◽  
Vol Accepted ◽  
Author(s):  
Mohammad Hamoda
Keyword(s):  
Primary Submodules

scholarly journals On intuitionistic L-fuzzy primary and P-primary submodules

2020 ◽  
Vol 8 (4) ◽  
pp. 1417-1426
Author(s):  
Sharma P. K. ◽  
Kanchan K
Keyword(s):  
Primary Submodules

Hopf Algebras from Branching Rules

1983 ◽  
Vol 35 (1) ◽  
pp. 177-192 ◽  
Author(s):  
P. Hoffman
Keyword(s):  
Abelian Group ◽  
Symmetric Group ◽  
Hopf Algebras ◽  
Homogeneous Element ◽  
Algebra Structure ◽  
Graded Algebra ◽  
Finitely Generated ◽  
Dimension Zero ◽  
Branching Rules

Below we work out the algebra structure of some Hopf algebras which arise concretely in restricting representations of the symmetric group to certain subgroups. The basic idea generalizes that used by Adams [1] for H*(BSU). The question arose in discussions with H. K. Farahat. I would like to thank him for his interest in the work and to acknowledge the usefulness of several stimulating conversations with him.1. Review and statement of results. A homogeneous element of a graded abelian group will have its gradation referred to as its dimension. In all such groups below there will be no non-zero elements with negative or odd dimension. A graded algebra (resp. coalgebra) will be associative (resp. coassociative), strictly commutative (resp. co-commutative) and in dimension zero will be isomorphic to the ground ring F, providing the unit (resp. counit). We shall deal amost entirely with F = Z or F = Z/p for a prime p; the cases F = 0 or a localization of Z will occur briefly. In every case, the component in each dimension will be a finitely generated free F-module, so dualization works simply.

scholarly journals In Situ Formation of a Metastable β-Ti Alloy by Laser Powder Bed Fusion (L-PBF) of Vanadium and Iron Modified Ti-6Al-4V

Metals ◽  
2018 ◽  
Vol 8 (12) ◽  
pp. 1067 ◽  
Author(s):  
Florian Huber ◽  
Thomas Papke ◽  
Christian Scheitler ◽  
Lukas Hanrieder ◽  
Marion Merklein ◽  
...  
Keyword(s):  
Mechanical Properties ◽  
Heat Treatment ◽  
Homogeneous Element ◽  
Alloy Formation ◽  
Powder Bed Fusion ◽  
Laser Powder Bed Fusion ◽  
Powder Bed ◽  
Β Phase ◽  
Stabilizing Effect

The aim of this work is to investigate the β-Ti-phase-stabilizing effect of vanadium and iron added to Ti-6Al-4V powder by means of heterogeneous powder mixtures and in situ alloy-formation during laser powder bed fusion (L-PBF). The resulting microstructure was analyzed by metallographic methods, scanning electron microscopy (SEM), and electron backscatter diffraction (EBSD). The mechanical properties were characterized by compression tests, both prior to and after heat-treating. Energy dispersive X-ray spectroscopy showed a homogeneous element distribution, proving the feasibility of in situ alloying by LPBF. Due to the β-phase-stabilizing effect of V and Fe added to Ti-6Al-4V, instead of an α’-martensitic microstructure, an α/β-microstructure containing at least 63.8% β-phase develops. Depending on the post L-PBF heat-treatment, either an increased upsetting at failure (33.9%) compared to unmodified Ti-6Al-4V (28.8%), or an exceptional high compressive yield strength (1857 ± 35 MPa compared to 1100 MPa) were measured. The hardness of the in situ alloyed material ranges from 336 ± 7 HV0.5, in as-built condition, to 543 ± 13 HV0.5 after precipitation-hardening. Hence, the range of achievable mechanical properties in dependence of the post-L-PBF heat-treatment can be significantly expanded in comparison to unmodified Ti-6Al-4V, thus providing increased flexibility for additive manufacturing of titanium parts.

scholarly journals Graded Primary Submodules over Multiplication Modules

2017 ◽  
Vol 225 ◽  
pp. 012079
Author(s):  
Malik Bataineh ◽  
Ala’ Lutfi Khazaaleh
Keyword(s):  
Primary Submodules

scholarly journals Architecture of Jesuit Churches in the Former Polish-Lithuanian Commonwealth, 1564–1773

2018 ◽  
Vol 5 (3) ◽  
pp. 352-384
Author(s):  
Andrzej Betlej
Keyword(s):  
Eighteenth Century ◽  
Homogeneous Element ◽  
The Third ◽  
Late Eighteenth Century ◽  
Late Eighteenth ◽  
The Subject ◽  
Relative Decline

The article presents the history and accomplishments of Jesuit architecture in the Polish-Lithuanian Commonwealth from the late sixteenth to the late eighteenth century. The author sees Jesuit architecture as a distinct and homogeneous element within Polish architecture. The paper starts with a brief presentation of the existing research in the subject. It moves on to enumerate the activities of the Society in the field of construction, divided into three major booms: the first roughly between 1575 and 1650, the second between 1670 and 1700, and the third from 1740 to 1770, divided by periods of relative decline caused by a succession of devastating wars. The paper identifies the most important architects involved in the construction of Jesuit churches, as well as their most notable works. The paper ends with a brief note concerning the fate of the Jesuit churches after the suppression of the Society and the partitions of Poland.

scholarly journals On generalizations of classical primary submodules over commutative rings

2018 ◽  
Vol 5 (1) ◽  
pp. 1458556
Author(s):  
P. Yiarayong ◽  
M.  ◽  
Hari M. Srivastava
Keyword(s):  
Commutative Rings ◽  
Primary Submodules

scholarly journals On Generalizations of phi-2-Absorbing Primary Submodules

2018 ◽  
Vol 11 (1) ◽  
pp. 35
Author(s):  
Pairote Yiarayong ◽  
Manoj Siripitukdet
Keyword(s):  
Positive Integer ◽  
Sufficient Conditions ◽  
Commutative Rings ◽  
Necessary And Sufficient Conditions ◽  
Primary Submodules ◽  
Primary Submodule ◽  
Necessary And Sufficient

Let $\phi: S(M) \rightarrow S(M) \cup \left\lbrace \emptyset\right\rbrace $ be a function where $S(M)$ is the set of all submodules of $M$. In this paper, we extend the concept of $\phi$-$2$-absorbing primary submodules to the context of $\phi$-$2$-absorbing semi-primary submodules. A proper submodule $N$ of $M$ is called a $\phi$-$2$-absorbing semi-primary submodule, if for each $m \in M$ and $a_{1}, a_{2}\in R$ with $a_{1}a_{2}m \in N - \phi(N)$, then $a_{1}a_{2}\in \sqrt{(N : M)}$ or  $a_{1}m \in N$ or $a^{n}_{2}m\in N$, for some positive integer $n$. Those are extended from $2$-absorbing primary, weakly $2$-absorbing primary, almost $2$-absorbing primary, $\phi_{n}$-$2$-absorbing primary, $\omega$-$2$-absorbing primary and $\phi$-$2$-absorbing primary submodules, respectively. Some characterizations of $2$-absorbing semi-primary, $\phi_{n}$-$2$-absorbing semi-primary and $\phi$-$2$-absorbing semi-primary submodules are obtained. Moreover, we investigate relationships between $2$-absorbing semi-primary, $\phi_{n}$-$2$-absorbing semi-primary and $\phi$-primary submodules of modules over commutative rings. Finally, we obtain necessary and sufficient conditions of a $\phi$-$\phi$-$2$-absorbing semi-primary in order to be a $\phi$-$2$-absorbing semi-primary.

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