STRONGLY SEMIPRIME AND STRONGLY NILPOTENT IDEALS

2016 ◽  
Vol 38 (1) ◽  
pp. 39-43
Author(s):  
Ahmad Khaksari
Keyword(s):  
Strongly Nilpotent

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.

On generalized strongly nilpotent matrices

1996 ◽  
Vol 41 (1) ◽  
pp. 19-22 ◽  
Author(s):  
Jie-Tai Yu ◽  
H. Bass ◽  
E. H. Connell ◽  
D. Wright
Keyword(s):  
Nilpotent Matrices ◽  
Strongly Nilpotent

A note on the generalized nilpotent elements of a module

2020 ◽  
pp. 289-294
Author(s):  
Saugata Purkayastha ◽  
Helen K. Saikia
Keyword(s):  
Nilpotent Element ◽  
Zero Element ◽  
Nilpotent Elements ◽  
Strongly Nilpotent

In this paper, we introduce the notion of the generalized nilpotent element of a module. In \cite{Groenewald}, the notion of nilpotent element of a module is introduced in the following sense: a non-zero element $m$ of an $R$-module $M$ is said to be nilpotent if there exists some $a\in R$ such that $a^k m=0$ but $am\neq 0$ for some $k\in \mathbb N$. In our present work we aim to generalize this notion. We have extended this notion to the strongly nilpotent element of a module.

RESIDUAL SMALLNESS AND WEAK CENTRALITY

2003 ◽  
Vol 13 (01) ◽  
pp. 35-59 ◽  
Author(s):  
KEITH A. KEARNES ◽  
EMIL W. KISS
Keyword(s):  
Finite Number ◽  
Abelian Variety ◽  
Nilpotent Algebra ◽  
Finite Algebras ◽  
Small Variety ◽  
Special Cases ◽  
Strongly Nilpotent ◽  
Nilpotent Algebras ◽  
Minimal Algebra

We develop a method of creating skew congruences on subpowers of finite algebras using groups of twin polynomials, and apply it to the investigation of residually small varieties generated by nilpotent algebras. We prove that a residually small variety generated by a finite nilpotent (in particular, a solvable E-minimal) algebra is weakly abelian. Conversely, we show in two special cases that a weakly abelian variety is residually bounded by a finite number: when it is generated by an E-minimal, or by a finite strongly nilpotent algebra. This establishes the RS-conjecture for E-minimal algebras.

On the Strong Nilpotence Problem

2014 ◽  
Vol 21 (01) ◽  
pp. 117-128 ◽  
Author(s):  
Xiaosong Sun
Keyword(s):  
Homogeneous Polynomial ◽  
Polynomial Maps ◽  
Strongly Nilpotent

In this paper, we describe the structure of quadratic homogeneous polynomial maps F=X+H with JH3=0. As a consequence we show that in dimension n ≤ 6, JH is strongly nilpotent, or equivalently F=X+H is linearly triangularizable.

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.

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