Characterization of non-nilpotent elements of the Z-module Z/(p_1^{k_1} x...x p_n^{k_n})Z

2013 ◽  
Vol 7 ◽  
pp. 699-702
Author(s):  
David Ssevviiri
Keyword(s):  
Nilpotent Elements

scholarly journals On strong CNZ rings and their extensions

2020 ◽  
Vol 9 (2) ◽  
pp. 80-92
Author(s):  
Chenar Abdul Kareem Ahmed
Keyword(s):  
Nilpotent Elements ◽  
Left And Right

T.K. Kwak and Y. Lee called a ring R satisfy the commutativity of nilpotent elements at zero[1] if ab = 0 for a, b ∈ N(R) implies ba = 0. For simplicity, a ring R is called CNZ if it satisfies the commutativity of nilpotent elements at zero. In this paper we study an extension of a CNZ ring with its endomorphism. An endomorphism α of a ring R is called strong right ( resp., left) CNZ if whenever aα(b) = 0(resp., α(a)b = 0 ) for a, b ∈ N(R) ba = 0. A ring R is called strong right (resp., left) α-CNZ if there exists a strong right (resp., left) CNZ endomorphism α of R, and the ring R is called strong α- CNZ if R is both strong left and right α- CNZ. Characterization of strong α- CNZ rings and their related properties including extensions are investigated . In particular, it’s shown that a ring R is reduced if and only if U2(R) is a CNZ ring. Furthermore extensions of strong α- CNZ rings are studied.

scholarly journals Nilpotents and units in skew polynomial rings over commutative rings

1979 ◽  
Vol 28 (4) ◽  
pp. 423-426 ◽  
Author(s):  
M. Rimmer ◽  
K. R. Pearson
Keyword(s):  
Commutative Ring ◽  
Finite Order ◽  
Polynomial Ring ◽  
Commutative Rings ◽  
Polynomial Rings ◽  
Skew Polynomial Ring ◽  
Nilpotent Elements ◽  
Skew Polynomial Rings ◽  
Skew Polynomial

AbstractLet R be a commutative ring with an automorphism ∞ of finite order n. An element f of the skew polynomial ring R[x, α] is nilpotent if and only if all coefficients of fn are nilpotent. (The case n = 1 is the well-known description of the nilpotent elements of the ordinary polynomial ring R[x].) A characterization of the units in R[x, α] is also given.

scholarly journals On Duo Rings

1960 ◽  
Vol 3 (2) ◽  
pp. 167-172 ◽  
Author(s):  
G. Thierrin
Keyword(s):  
Commutative Rings ◽  
Prime Ideals ◽  
Subdirectly Irreducible ◽  
Nilpotent Elements

Following E. H. Feller [l], a ring R is called a duo ring if every one-sided ideal of R is a two-sided ideal.In the first part of this paper, we give some properties of duo rings and we show that the set of the nilpotent elements of a duo ring R is an ideal, the intersection of the completely prime ideals of R.It is easy to see that every duo ring is a subdirect sum of subdirectly irreducible duo rings. We give in the second part of this paper a characterization of the subdirectly irreducible duo rings. This characterization is quite similar to the characterization of the subdirectly irreducible commutative rings, due to N. H. McCoy [2], whose methods we use.

scholarly journals Natural Filtrations of Infinite-Dimensional Modular Contact Superalgebras

2014 ◽  
Vol 2014 ◽  
pp. 1-9
Author(s):  
Qiang Mu
Keyword(s):  
Positive Characteristic ◽  
Lie Superalgebras ◽  
Closed Field ◽  
Infinite Dimensional ◽  
Intrinsic Characterization ◽  
Nilpotent Elements ◽  
Natural Filtration ◽  
Field Of Positive Characteristic

The natural filtration of the infinite-dimensional contact superalgebra over an algebraic closed field of positive characteristic is proved to be invariant under automorphisms by characterizing ad-nilpotent elements and the subalgebras generated by certain ad-nilpotent elements. Moreover, we obtain an intrinsic characterization of contact superalgebras and a property of automorphisms of these Lie superalgebras.

scholarly journals Одно замечание о периодических кольцах

2021 ◽  
pp. 109-111
Author(s):  
P.V. Danchev
Keyword(s):  
Arbitrary Ring ◽  
Nilpotent Elements ◽  
Periodic Rings ◽  
The Difference ◽  
Surprising Fact

We obtain a new and non-trivial characterization of periodic rings (that are those rings $R$ for which, for each element $x$ in $R$, there exists two different integers $m$, $n$ strictly greater than $1$ with the property $x^m=x^n$) in terms of nilpotent elements which supplies recent results in this subject by Cui--Danchev published in (J. Algebra \& Appl., 2020) and by Abyzov--Tapkin published in (J. Algebra \& Appl., 2022). Concretely, we state and prove the slightly surprising fact that an arbitrary ring $R$ is periodic if, and only if, for every element~$x$ from $R$, there are integers $m>1$ and $n>1$ with $m\not= n$ such that the difference $x^m-x^n$ is a nilpotent.

Associative rings in which nilpotents form an ideal

2019 ◽  
Vol 56 (2) ◽  
pp. 177-184
Author(s):  
Orest D. Artemovych
Keyword(s):  
Unit Group ◽  
Lie Ideal ◽  
Nilpotent Elements ◽  
Associative Rings ◽  
Torsion Free

Abstract It is shown that if N(R) is a Lie ideal of R (respectively Jordan ideal and R is 2-torsion-free), then N(R) is an ideal. Also, it is presented a characterization of Noetherian NR rings with central idempotents (respectively with the commutative set of nilpotent elements, the Abelian unit group, the commutative commutator set).

Rings with No Nilpotent Elements and with the Maximum Condition on Annihilators

1974 ◽  
Vol 17 (1) ◽  
pp. 35-38 ◽  
Author(s):  
W. H. Cornish ◽  
P. N. Stewart
Keyword(s):  
Maximum Condition ◽  
Annihilator Ideal ◽  
Nilpotent Elements ◽  
Reduced Ring ◽  
Minimal Prime

Rings (all of which are assumed to be associative) with no non-zero nilpotent elements will be called reduced rings; R is a reduced ring if and only if x2=0 implies x=0, for all x∈R. In 2. we prove that the following conditions on an annihilator ideal I of a reduced ring are equivalent: I is a maximal annihilator, I is prime, I is a minimal prime, I is completely prime. A characterization of reduced rings with the maximum condition on annihilators is given in 3.

Infinite-dimensional Modular Lie Superalgebras W and S of Cartan Type

2006 ◽  
Vol 13 (02) ◽  
pp. 197-210 ◽  
Author(s):  
Yongzheng Zhang ◽  
Wende Liu
Keyword(s):  
Automorphism Groups ◽  
Lie Superalgebras ◽  
Infinite Dimensional ◽  
Cartan Type ◽  
Intrinsic Characterization ◽  
Nilpotent Elements ◽  
Modular Lie Superalgebras

The natural filtrations of infinite-dimensional modular Lie superalgebras W and S are proved to be invariant under their automorphism groups by means of investigating the ad-nilpotent elements and determining certain subalgebras generated by ad-nilpotent elements. As an application, we obtain an intrinsic characterization of W and S, and give a property of the automorphisms of these modular Lie superalgebras.

Morphological Characterization of Mycobacteriophage R1

1974 ◽  
Vol 32 ◽  
pp. 254-255
Author(s):  
B. L. Soloff ◽  
T. A. Rado
Keyword(s):  
Carbon Source ◽  
Stationary Phase ◽  
Growth Phase ◽  
Minimal Medium ◽  
Morphological Characterization ◽  
Logarithmic Growth Phase ◽  
Complete Lysis ◽  
Lysogenic Culture ◽  
Logarithmic Growth

Mycobacteriophage R1 was originally isolated from a lysogenic culture of M. butyricum. The virus was propagated on a leucine-requiring derivative of M. smegmatis, 607 leu−, isolated by nitrosoguanidine mutagenesis of typestrain ATCC 607. Growth was accomplished in a minimal medium containing glycerol and glucose as carbon source and enriched by the addition of 80 μg/ ml L-leucine. Bacteria in early logarithmic growth phase were infected with virus at a multiplicity of 5, and incubated with aeration for 8 hours. The partially lysed suspension was diluted 1:10 in growth medium and incubated for a further 8 hours. This permitted stationary phase cells to re-enter logarithmic growth and resulted in complete lysis of the culture.

AEM analysis of crystallization in Ti-based amorphous alloys

1983 ◽  
Vol 41 ◽  
pp. 270-271
Author(s):  
A.R. Pelton ◽  
A.F. Marshall ◽  
Y.S. Lee
Keyword(s):  
Magnetic Properties ◽  
Amorphous Alloys ◽  
Amorphous Materials ◽  
Isothermal Annealing ◽  
Amorphous Matrix ◽  
Nucleation And Growth ◽  
Current Interest ◽  
Amorphous Films ◽  
Electrical And Magnetic Properties

Amorphous materials are of current interest due to their desirable mechanical, electrical and magnetic properties. Furthermore, crystallizing amorphous alloys provides an avenue for discerning sequential and competitive phases thus allowing access to otherwise inaccessible crystalline structures. Previous studies have shown the benefits of using AEM to determine crystal structures and compositions of partially crystallized alloys. The present paper will discuss the AEM characterization of crystallized Cu-Ti and Ni-Ti amorphous films.Cu60Ti40: The amorphous alloy Cu60Ti40, when continuously heated, forms a simple intermediate, macrocrystalline phase which then transforms to the ordered, equilibrium Cu3Ti2 phase. However, contrary to what one would expect from kinetic considerations, isothermal annealing below the isochronal crystallization temperature results in direct nucleation and growth of Cu3Ti2 from the amorphous matrix.

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