Some characteristics of cyclic prime, weakly prime and almost prime submodule of Gaussian integer modulo over integer

2021 ◽  
Author(s):  
Rina Juliana ◽  
I. Gede Adhitya Wisnu Wardhana ◽  
Irwansyah
Keyword(s):  
Gaussian Integer ◽  
Prime Submodule ◽  
Almost Prime

scholarly journals A note on almost prime submodule of CSM module over principal ideal domain

2021 ◽  
Vol 2106 (1) ◽  
pp. 012011
Author(s):  
I G A W Wardhana ◽  
N D H Nghiem ◽  
N W Switrayni ◽  
Q Aini
Keyword(s):  
Prime Ideal ◽  
Algebraic Structure ◽  
Principal Ideal ◽  
Ring Theory ◽  
Principal Ideal Domain ◽  
Module Theory ◽  
Ideal Domain ◽  
Multiplication Operation ◽  
Prime Submodule ◽  
Almost Prime

Abstract An almost prime submodule is a generalization of prime submodule introduced in 2011 by Khashan. This algebraic structure was brought from an algebraic structure in ring theory, prime ideal, and almost prime ideal. This paper aims to construct similar properties of prime ideal and almost prime ideal from ring theory to module theory. The problem that we want to eliminate is the multiplication operation, which is missing in module theory. We use the definition of module annihilator to bridge the gap. This article gives some properties of the prime submodule and almost prime submodule of CMS module over a principal ideal domain. A CSM module is a module that every cyclic submodule. One of the results is that the idempotent submodule is an almost prime submodule.

Study of Safe Elliptic Curve Cryptography over Gaussian Integer

2020 ◽  
Vol E103.A (12) ◽  
pp. 1624-1628
Author(s):  
Kazuki NAGANUMA ◽  
Takashi SUZUKI ◽  
Hiroyuki TSUJI ◽  
Tomoaki KIMURA
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curve Cryptography ◽  
Gaussian Integer

Perfect Gaussian Integer Sequence Pairs

2014 ◽  
Vol E97.A (7) ◽  
pp. 1568-1575 ◽  
Author(s):  
Chengqian XU ◽  
Xiuping PENG ◽  
Kai LIU
Keyword(s):  
Gaussian Integer ◽  
Integer Sequence

Sifat-sifat Submodul Prima dan Submodul Prima Lemah

2020 ◽  
Vol 2 (2) ◽  
pp. 183
Author(s):  
Hisyam Ihsan ◽  
Muhammad Abdy ◽  
Samsu Alam B
Keyword(s):  
Prime Ideal ◽  
Literature Study ◽  
Prime Submodules ◽  
Unitary Module ◽  
Prime Submodule ◽  
Definition Of ◽  
The Relationship

Penelitian ini merupakan penelitian kajian pustaka yang bertujuan untuk mengkaji sifat-sifat submodul prima dan submodul prima lemah serta hubungan antara keduanya. Kajian dimulai dari definisi submodul prima dan submodul prima lemah, selanjutnya dikaji mengenai sifat-sifat dari keduanya. Pada penelitian ini, semua ring yang diberikan adalah ring komutatif dengan unsur kesatuan dan modul yang diberikan adalah modul uniter. Sebagai hasil dari penelitian ini diperoleh beberapa pernyataan yang ekuivalen, misalkan  suatu -modul ,  submodul sejati di  dan ideal di , maka ketiga pernyataan berikut ekuivalen, (1)  merupakan submodul prima, (2) Setiap submodul tak nol dari   -modul memiliki annihilator yang sama, (3) Untuk setiap submodul  di , subring  di , jika berlaku  maka  atau . Di lain hal, pada submodul prima lemah jika diberikan  suatu -modul,  submodul sejati di , maka pernyataan berikut ekuivalen, yaitu (1) Submodul  merupakan submodul prima lemah, (2) Untuk setiap , jika  maka . Selain itu, didapatkan pula hubungan antara keduanya, yaitu setiap submodul prima merupakan submodul prima lemah.Kata Kunci: Submodul Prima, Submodul Prima Lemah, Ideal Prima. This research is literature study that aims to examine the properties of prime submodules and weakly prime submodules and the relationship between  both of them. The study starts from the definition of prime submodules and weakly prime submodules, then reviewed about the properties both of them. Throughout this paper all rings are commutative with identity and all modules are unitary. As the result of this research, obtained several equivalent statements, let  be a -module,  be a proper submodule of  and  ideal of , then the following three statetments are equivalent, (1)  is a prime submodule, (2) Every nonzero submodule of   -module has the same annihilator, (3) For any submodule  of , subring  of , if  then  or . In other case, for weakly prime submodules, if given  is a unitary -module,  be a proper submodule of , then the following statements are equivalent, (1)  is a weakly prime submodule, (2) For any , if  then . In addition, also found the relationship between both of them, i.e. any prime submodule is weakly prime submodule.Keywords: Prime Submodules, Weakly Prime Submdules, Prime Ideal.

scholarly journals BOUNDED AND FULLY BOUNDED MODULES

2011 ◽  
Vol 84 (3) ◽  
pp. 433-440
Author(s):  
A. HAGHANY ◽  
M. MAZROOEI ◽  
M. R. VEDADI
Keyword(s):  
Krull Dimension ◽  
Finitely Generated ◽  
Morita Invariant ◽  
Prime Submodule ◽  
Invariant Properties ◽  
Noetherian Modules

AbstractGeneralizing the concept of right bounded rings, a module MR is called bounded if annR(M/N)≤eRR for all N≤eMR. The module MR is called fully bounded if (M/P) is bounded as a module over R/annR(M/P) for any ℒ2-prime submodule P◃MR. Boundedness and right boundedness are Morita invariant properties. Rings with all modules (fully) bounded are characterized, and it is proved that a ring R is right Artinian if and only if RR has Krull dimension, all R-modules are fully bounded and ideals of R are finitely generated as right ideals. For certain fully bounded ℒ2-Noetherian modules MR, it is shown that the Krull dimension of MR is at most equal to the classical Krull dimension of R when both dimensions exist.

scholarly journals Gaussian Integers

2013 ◽  
Vol 21 (2) ◽  
pp. 115-125
Author(s):  
Yuichi Futa ◽  
Hiroyuki Okazaki ◽  
Daichi Mizushima ◽  
Yasunari Shidama
Keyword(s):  
Number Field ◽  
Rational Number ◽  
Rational Numbers ◽  
Gaussian Integer ◽  
Algebraic Integers ◽  
Quotient Field ◽  
Gaussian Integers ◽  
Integer Ring

Summary Gaussian integer is one of basic algebraic integers. In this article we formalize some definitions about Gaussian integers [27]. We also formalize ring (called Gaussian integer ring), Z-module and Z-algebra generated by Gaussian integer mentioned above. Moreover, we formalize some definitions about Gaussian rational numbers and Gaussian rational number field. Then we prove that the Gaussian rational number field and a quotient field of the Gaussian integer ring are isomorphic.

scholarly journals On a generalization of prime submodules of a module over a commutative ring

2017 ◽  
Vol 37 (1) ◽  
pp. 153-168
Author(s):  
Hosein Fazaeli Moghimi ◽  
Batool Zarei Jalal Abadi
Keyword(s):  
Commutative Ring ◽  
Dedekind Domain ◽  
Finitely Generated ◽  
Multiplication Module ◽  
Maximal Ideals ◽  
Prime Submodules ◽  
Prime Submodule

‎Let $R$ be a commutative ring with identity‎, ‎and $n\geq 1$ an integer‎. ‎A proper submodule $N$ of an $R$-module $M$ is called‎ ‎an $n$-prime submodule if whenever $a_1 \cdots a_{n+1}m\in N$ for some non-units $a_1‎, ‎\ldots‎ , ‎a_{n+1}\in R$ and $m\in M$‎, ‎then $m\in N$ or there are $n$ of the $a_i$'s whose product is in $(N:M)$‎. ‎In this paper‎, ‎we study $n$-prime submodules as a generalization of prime submodules‎. ‎Among other results‎, ‎it is shown that if $M$ is a finitely generated faithful multiplication module over a Dedekind domain $R$‎, ‎then every $n$-prime submodule of $M$ has the form $m_1\cdots m_t M$ for some maximal ideals $m_1,\ldots,m_t$ of $R$ with $1\leq t\leq n$‎.

Strongly 0-dimensional Modules

2014 ◽  
Vol 57 (1) ◽  
pp. 159-165 ◽  
Author(s):  
Kürşat Hakan Oral ◽  
Neslihan Ayşen Özkirişci ◽  
Ünsal Tekir
Keyword(s):  
Multiplication Module ◽  
Finite Intersection ◽  
Von Neumann ◽  
Prime Submodules ◽  
Von Neumann Regular ◽  
Prime Submodule ◽  
Dimensional Module

AbstractIn a multiplication module, prime submodules have the following property: if a prime submodule contains a finite intersection of submodules, then one of the submodules is contained in the prime submodule. In this paper, we generalize this property to infinite intersection of submodules and call such prime submodules strongly prime submodules. A multiplication module in which every prime submodule is strongly prime will be called a strongly 0-dimensional module. It is also an extension of strongly 0-dimensional rings. After this we investigate properties of strongly 0-dimensional modules and give relations of von Neumann regular modules, Q-modules and strongly 0-dimensional modules

scholarly journals Reductions of an elliptic curve with almost prime orders

2005 ◽  
Vol 119 (3) ◽  
pp. 265-289 ◽  
Author(s):  
Alina Carmen Cojocaru
Keyword(s):  
Elliptic Curve ◽  
Almost Prime
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