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.

The set of strong torsion elements of a module over a noncommutative ring

2019 ◽  
Vol 19 (07) ◽  
pp. 2050123
Author(s):  
A. Farzi-safarabadi ◽  
R. Beyranvand
Keyword(s):  
Nonzero Ideal ◽  
Noncommutative Ring ◽  
Arbitrary Ring ◽  
Ideal Formula ◽  
Prime Submodules

Let [Formula: see text] be an arbitrary ring and [Formula: see text] be a nonzero right [Formula: see text]-module. In this paper, we introduce the set [Formula: see text], for some nonzero ideal [Formula: see text] of [Formula: see text] of strong torsion elements of [Formula: see text] and the properties of this set are investigated. In particular, we are interested when [Formula: see text] is a submodule of [Formula: see text] and when it is a union of prime submodules of [Formula: see text].

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 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

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.

On almost prime submodules

2012 ◽  
Vol 32 (2) ◽  
pp. 645-651 ◽  
Author(s):  
Hani A. Khashan
Keyword(s):  
Prime Submodules ◽  
Almost Prime

The Zariski Topology on the Prime Spectrum of a Module over Noncommutative Rings

2009 ◽  
Vol 16 (04) ◽  
pp. 691-698 ◽  
Author(s):  
Ünsal Tekir
Keyword(s):  
Associative Ring ◽  
Zariski Topology ◽  
Prime Spectrum ◽  
Noncommutative Rings ◽  
Prime Submodules

Let R be an associative ring with identity and M an R-module. Let Spec (M) be the set of all prime submodules of M. We topologize Spec (M) with the Zariski topology and prove some useful 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 $M$.

scholarly journals On radical formula in modules over noncommutative rings

Filomat ◽  
2020 ◽  
Vol 34 (2) ◽  
pp. 443-449
Author(s):  
Ortac Öneş
Keyword(s):  
Prime Ideals ◽  
Direct Connection ◽  
Noncommutative Rings ◽  
Nilpotent Elements ◽  
Prime Submodules ◽  
Prime Submodule ◽  
Strongly Nilpotent

This paper examines the radical formula in noncommutative case and for this purpose, a generalization of prime submodule is defined. It is proved that there is a direct connection between onesided prime ideals and one-sided prime submodules. Moreover the connections between the intersection of all one-sided prime submodules and strongly nilpotent elements of a module are studied.

ON ALMOST PRIME SUBMODULES OF A MODULE OVER A PRINCIPAL IDEAL DOMAIN

2016 ◽  
Vol 38 (2) ◽  
pp. 121-128
Author(s):  
I Gede Adhitya Wisnu Wardhana ◽  
Pudji Astuti ◽  
Intan Muchtadi-Alamsyah
Keyword(s):  
Principal Ideal ◽  
Principal Ideal Domain ◽  
Ideal Domain ◽  
Prime Submodules ◽  
Almost Prime
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