scholarly journals PRIME M-IDEALS, M-PRIME SUBMODULES, M-PRIME RADICAL AND M-BAER'S LOWER NILRADICAL OF MODULES

2013 ◽  
Vol 50 (6) ◽  
pp. 1271-1290
Author(s):  
John A. Beachy ◽  
Mahmood Behboodi ◽  
Faezeh Yazdi
Keyword(s):  
Prime Radical ◽  
Prime Submodules

A GENERALIZATION OF BAER'S LOWER NILRADICAL FOR MODULES

2007 ◽  
Vol 06 (02) ◽  
pp. 337-353 ◽  
Author(s):  
MAHMOOD BEHBOODI
Keyword(s):  
Prime Ring ◽  
Artinian Ring ◽  
Prime Radical ◽  
Nilpotent Elements ◽  
Noetherian Domain ◽  
Prime Submodules ◽  
Strongly Nilpotent ◽  
Commutative Domain ◽  
Torsion Modules

Let M be a left R-module. A proper submodule P of M is called classical prime if for all ideals [Formula: see text] and for all submodules N ⊆ M, [Formula: see text] implies that [Formula: see text] or [Formula: see text]. We generalize the Baer–McCoy radical (or classical prime radical) for a module [denoted by cl.rad R(M)] and Baer's lower nilradical for a module [denoted by Nil *(RM)]. For a module RM, cl.rad R(M) is defined to be the intersection of all classical prime submodules of M and Nil *(RM) is defined to be the set of all strongly nilpotent elements of M (defined later). It is shown that, for any projective R-module M, cl.rad R(M) = Nil *(RM) and, for any module M over a left Artinian ring R, cl.rad R(M) = Nil *(RM) = Rad (M) = Jac (R)M. In particular, if R is a commutative Noetherian domain with dim (R) ≤ 1, then for any module M, we have cl.rad R(M) = Nil *(RM). We show that over a left bounded prime left Goldie ring, the study of Baer–McCoy radicals of general modules reduces to that of torsion modules. Moreover, over an FBN prime ring R with dim (R) ≤ 1 (or over a commutative domain R with dim (R) ≤ 1), every semiprime submodule of any module is an intersection of classical prime submodules.

scholarly journals Approximaitly Prime Submodules and Some Related Concepts

2019 ◽  
Vol 32 (2) ◽  
pp. 103
Author(s):  
Ali Sh. Ajeel ◽  
Haibat K. Mohammad Ali
Keyword(s):  
Prime Ideal ◽  
Commutative Ring ◽  
Research Note ◽  
Prime Radical ◽  
Basic Properties ◽  
Prime Submodules ◽  
Prime Submodule ◽  
Definition Of

In this research note approximately prime submodules is defined as a new generalization of prime submodules of unitary modules over a commutative ring with identity. A proper submodule  of an -module  is called an approximaitly prime submodule of  (for short app-prime submodule), if when ever , where , , implies that either  or . So, an ideal  of a ring  is called app-prime ideal of  if   is an app-prime submodule of -module . Several basic properties, characterizations and examples of approximaitly prime submodules were given. Furthermore, the definition of approximaitly prime radical of submodules of modules were introduced, and some of it is properties were established.

scholarly journals Primary modules over commutative rings

2001 ◽  
Vol 43 (1) ◽  
pp. 103-111 ◽  
Author(s):  
Patrick F. Smith
Keyword(s):  
Commutative Ring ◽  
Commutative Rings ◽  
Prime Radical ◽  
Finitely Generated ◽  
One Dimensional ◽  
Prime Submodules ◽  
Primary Submodules ◽  
Commutative Domain

The radical of a module over a commutative ring is the intersection of all prime submodules. It is proved that if R is a commutative domain which is either Noetherian or a UFD then R is one-dimensional if and only if every (finitely generated) primary R-module has prime radical, and this holds precisely when every (finitely generated) R-module satisfies the radical formula for primary submodules.

Almost prime and weakly prime submodules

2019 ◽  
Vol 18 (07) ◽  
pp. 1950129 ◽  
Author(s):  
P. Karimi Beiranvand ◽  
R. Beyranvand
Keyword(s):  
Prime Radical ◽  
Noncommutative Rings ◽  
Arbitrary Ring ◽  
Ideal Formula ◽  
Prime Submodules ◽  
Almost Prime

Let [Formula: see text] be an arbitrary ring and [Formula: see text] be a right [Formula: see text]-module. A proper submodule [Formula: see text] of [Formula: see text] is called almost prime (respectively, weakly prime) if for each submodule [Formula: see text] of [Formula: see text] and each ideal [Formula: see text] of [Formula: see text] that [Formula: see text] and [Formula: see text] (respectively, [Formula: see text]), then [Formula: see text] or [Formula: see text]. We study these notions which are new generalizations of the prime submodules over noncommutative rings and we obtain some related results. We show that these two concepts in some classes of modules coincide. Moreover, we investigate the conditions that [Formula: see text] is almost prime, where [Formula: see text] is a submodule of [Formula: see text] and [Formula: see text] is an ideal of [Formula: see text]. Also, the almost prime radical of modules will be introduced and we extend some known results.

scholarly journals On Fully Prime Radicals

2017 ◽  
Vol 23 (2) ◽  
pp. 33-45
Author(s):  
Indah Emilia Wijayanti ◽  
Dian Ariesta Yuwaningsih
Keyword(s):  
Prime Radical ◽  
Prime Submodules ◽  
Special Case

In this paper we give a further study on fully prime submodules. For any fully prime submodules we define a product called $\am$-product. The further investigation of fully prime submodules in this work, i.e. the fully m-system and fully prime radicals, is related to this product. We show that the fully prime radical of any submodules can be characterize by the fully m-system. As a special case, the fully prime radical of a module $M$ is the intersection of all minimal fully prime submodules of $M$.

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.

ON EXTENSION OF PRIME RADICAL IN 2-PRIMAL NEAR-RINGS

2020 ◽  
Vol 9 (3) ◽  
pp. 1339-1348
Author(s):  
B. Elavarasan ◽  
K. Porselvi and J. Catherine Grace John ◽  
Porselvi J. Catherine Grace John
Keyword(s):  
Prime Radical

Small Prime Modules and Small Prime Submodules

2012 ◽  
Vol 15 (4) ◽  
pp. 191-199
Author(s):  
Layla S. Mahmood ◽  
Keyword(s):  
Prime Submodules

Prime uniserial modules and rings

2019 ◽  
Vol 18 (02) ◽  
pp. 1950035 ◽  
Author(s):  
M. Behboodi ◽  
Z. Fazelpour
Keyword(s):  
Commutative Rings ◽  
Finitely Generated ◽  
Finitely Generated Module ◽  
Direct Sum ◽  
Dedekind Domains ◽  
Prime Submodules ◽  
Uniserial Modules ◽  
Structure Formula

We define prime uniserial modules as a generalization of uniserial modules. We say that an [Formula: see text]-module [Formula: see text] is prime uniserial ([Formula: see text]-uniserial) if its prime submodules are linearly ordered by inclusion, and we say that [Formula: see text] is prime serial ([Formula: see text]-serial) if it is a direct sum of [Formula: see text]-uniserial modules. The goal of this paper is to study [Formula: see text]-serial modules over commutative rings. First, we study the structure [Formula: see text]-serial modules over almost perfect domains and then we determine the structure of [Formula: see text]-serial modules over Dedekind domains. Moreover, we discuss the following natural questions: “Which rings have the property that every module is [Formula: see text]-serial?” and “Which rings have the property that every finitely generated module is [Formula: see text]-serial?”.

scholarly journals Projective modules and prime submodules

2006 ◽  
Vol 56 (2) ◽  
pp. 601-611 ◽  
Author(s):  
Mustafa Alkan ◽  
Yücel Tiraş
Keyword(s):  
Projective Modules ◽  
Prime Submodules
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