arXiv subjects

Shaban Ghalandarzadeh

Publications and source records attributed to Shaban Ghalandarzadeh.

8 recordsLinked to original sources

arXiv2023

$ϕ$-$δ$-$S$-primary submodule

In this paper, we introduce and investigate some properties of $ϕ$-$δ$-$S$-primary submodules, which is a generalization of the $ϕ$-$δ$-primary submodules and prime submodules in general. We extend a number of main results about $ϕ$-$δ$-primary submodule and prime submodule into the general framework. Moreover, we show that there is an one-to-one correspondence between the set of all $ϕ$-$δ$-$S$-primary submodules of an $R$-module $M$ containing $K$ and the set of all $ϕ$-$δ$-$S$-primary submodules of $\frac{M}{K}$.

math.AC↗

arXiv2022

On Radical of Intuitionistic Fuzzy Primary Submodule

In this paper, we further study the theory of Intuitionistic fuzzy submodules and we will define intuitionistic fuzzy primary submodule with the help of the definition of a radical submodule, and we also study the properties of these submodules. Furthermore, homomorphic image and pre-image of intuitionistic fuzzy primary submodule are investigated.

math.AC↗

arXiv2019

Solving Linear Systems over Tropical Semirings through Normalization Method and its Applications

In this paper, we introduce and analyze a normalization method for solving a system of linear equations over tropical semirings. We use a normalization method to construct an associated normalized matrix, which gives a technique for solving the system. If solutions exist, the method can also determine the degrees of freedom of the system. Moreover, we present a procedure to determine the column rank and the row rank of a matrix. Flowcharts for this normalization method and its applications are included as well.

math.AC↗

arXiv2019

On the Maximal Solution of A Linear System over Tropical Semirings

In this paper, we present methods for solving a system of linear equations, $ AX=b $, over tropical semirings. To this end, if possible, we first reduce the order of the system through some row-column analysis, and obtain a new system with fewer equations and variables. We then use the pseudo-inverse of the system matrix to solve the system if solutions exist. Moreover, we propose a new version of Cramer's rule to determine the maximal solution of the system. Maple procedures for computing the pseudo-inverse are included as well.

math.AC↗

arXiv2019

Solving Linear Systems over Idempotent Semifields through $LU$-factorization

In this paper, we introduce and analyze a new $LU$-factorization technique for square matrices over idempotent semifields. In particular, more emphasis is put on "max-plus" algebra here, but the work is extended to other idempotent semifields as well. We first determine the conditions under which a square matrix has $LU$ factors. Next, using this technique, we propose a method for solving square linear systems of equations whose system matrices are $LU$-factorizable. We also give conditions for an $LU$-factorizable system to have solutions. This work is an extension of similar techniques over fields. Maple procedures for this $LU$-factorization are also included.

math.AC↗

arXiv2019

Analysis of linear systems over idempotent semifields

In this paper, we present and analyze methods for solving a system of linear equations over idempotent semifields. The first method is based on the pseudo-inverse of the system matrix. We then present a specific version of Cramer's rule which is also related to the pseudo-inverse of the system matrix. In these two methods, the constant vector plays an implicit role in solvability of the system. Another method is called the normalization method in which both the system matrix and the constant vector play explicit roles in the solution process. Each of these methods yields the maximal solution if it exists. Finally, we show the maximal solutions obtained from these methods and some previous methods are all identical.

math.AC↗

arXiv2019

Multiplication Semimodules

Let $S$ be a semiring. An $S$-semimodule $M$ is called a multiplication semimodule if for each subsemimodule $N$ of $M$ there exists an ideal $I$ of $S$ such that $N=IM$. In this paper we investigate some properties of multiplication semimodules and generalize some results on multiplication modules to semimodules. We show that every multiplicatively cancellative multiplication semimodule is finitely generated and projective. Moreover, we characterize finitely generated cancellative multiplication $S$-semimodules when $S$ is a yoked semiring such that every maximal ideal of $S$ is subtractive.

math.AC↗

arXiv2017

Invertible Ideals and Gaussian Semirings

In the first section of this paper, we introduce the notions of fractional and invertible ideals of semirings and characterize invertible ideals of a semidomain. In section two, we define Prüfer semirings and characterize them in terms of valuation semirings. In this section, we also characterize Prüfer semirings in terms of some identities over its ideals such as $(I + J)(I \cap J) = IJ$ for all ideals $I$, $J$ of $S$. In the third section, we give a semiring version for the Gilmer-Tsang Theorem, which states that for a suitable family of semirings, the concepts of Prüfer and Gaussian semirings are equivalent. At last we end this paper by giving a plenty of examples of proper Gaussian and Prüfer semirings.

math.AC↗