On quasi-radical of near-ring of polynomials

2019 ◽  
Vol 56 (2) ◽  
pp. 252-259
Author(s):  
Ebrahim Hashemi ◽  
Fatemeh Shokuhifar ◽  
Abdollah Alhevaz
Keyword(s):  
Commutative Ring ◽  
Nilpotent Elements ◽  
Ring Of Polynomials

Abstract The intersection of all maximal right ideals of a near-ring N is called the quasi-radical of N. In this paper, first we show that the quasi-radical of the zero-symmetric near-ring of polynomials R0[x] equals to the set of all nilpotent elements of R0[x], when R is a commutative ring with Nil (R)2 = 0. Then we show that the quasi-radical of R0[x] is a subset of the intersection of all maximal left ideals of R0[x]. Also, we give an example to show that for some commutative ring R the quasi-radical of R0[x] coincides with the intersection of all maximal left ideals of R0[x]. Moreover, we prove that the quasi-radical of R0[x] is the greatest quasi-regular (right) ideal of it.

The Weakly Nilpotent Graph of a Commutative Ring

2017 ◽  
Vol 60 (2) ◽  
pp. 319-328
Author(s):  
Soheila Khojasteh ◽  
Mohammad Javad Nikmehr
Keyword(s):  
Commutative Ring ◽  
Chromatic Number ◽  
Disjoint Union ◽  
Independence Number ◽  
Artinian Ring ◽  
Clique Number ◽  
Commutative Rings ◽  
Unicyclic Graph ◽  
Nilpotent Elements ◽  
Vertex Set

AbstractLet R be a commutative ring with non-zero identity. In this paper, we introduce theweakly nilpotent graph of a commutative ring. The weakly nilpotent graph of R denoted by Γw(R) is a graph with the vertex set R* and two vertices x and y are adjacent if and only if x y ∊ N(R)*, where R* = R \ {0} and N(R)* is the set of all non-zero nilpotent elements of R. In this article, we determine the diameter of weakly nilpotent graph of an Artinian ring. We prove that if Γw(R) is a forest, then Γw(R) is a union of a star and some isolated vertices. We study the clique number, the chromatic number, and the independence number of Γw(R). Among other results, we show that for an Artinian ring R, Γw(R) is not a disjoint union of cycles or a unicyclic graph. For Artinan rings, we determine diam . Finally, we characterize all commutative rings R for which is a cycle, where is the complement of the weakly nilpotent graph of R.

Local multiplication maps on F[x]

2014 ◽  
Vol 14 (03) ◽  
pp. 1550029
Author(s):  
Kelly Aceves ◽  
Manfred Dugas
Keyword(s):  
Commutative Ring ◽  
Finite Fields ◽  
Linear Transformation ◽  
Ring Of Polynomials

Let F be a field and A a F-algebra. The F-linear transformation φ : A → A is a local multiplication map if for all a ∈ A there exists some ua ∈ A such that φ(a) = aua. Let [Formula: see text] denote the F-algebra of all local multiplication maps of A. If F is infinite and F[x] is the ring of polynomials over F, then it is known Lemma 1 in [J. Buckner and M. Dugas, Quasi-Localizations of ℤ, Israel J. Math.160 (2007) 349–370] that [Formula: see text]. The purpose of this paper is to study [Formula: see text] for finite fields F. It turns out that in this case [Formula: see text] is a "very" non-commutative ring of cardinality 2ℵ0 with many interesting properties.

scholarly journals Periodic rings with commuting nilpotents

1984 ◽  
Vol 7 (2) ◽  
pp. 403-406
Author(s):  
Hazar Abu-Khuzam ◽  
Adil Yaqub
Keyword(s):  
Positive Integer ◽  
Commutative Ring ◽  
Finite Fields ◽  
Boolean Ring ◽  
Fixed Integer ◽  
Integer Coefficients ◽  
Nilpotent Elements ◽  
Periodic Rings

LetRbe a ring (not necessarily with identity) and letNdenote the set of nilpotent elements ofR. Suppose that (i)Nis commutative, (ii) for everyxinR, there exists a positive integerk=k(x)and a polynomialf(λ)=fx(λ)with integer coefficients such thatxk=xk+1f(x), (iii) the setIn={x|xn=x}wherenis a fixed integer,n>1, is an ideal inR. ThenRis a subdirect sum of finite fields of at mostnelements and a nil commutative ring. This theorem, generalizes the “xn=x” theorem of Jacobson, and (takingn=2) also yields the well known structure of a Boolean ring. An Example is given which shows that this theorem need not be true if we merely assume thatInis a subring ofR.

On Functional Representations of a Ring without Nilpotent Elements

1971 ◽  
Vol 14 (3) ◽  
pp. 349-352 ◽  
Author(s):  
Kwangil Koh
Keyword(s):  
Prime Ideal ◽  
Commutative Ring ◽  
Ideal Space ◽  
Noncommutative Ring ◽  
Nilpotent Elements ◽  
Functional Representations

In [3, p. 149], J. Lambek gives a proof of a theorem, essentially due to Grothendieck and Dieudonne, that if R is a commutative ring with 1 then R is isomorphic to the ring of global sections of a sheaf over the prime ideal space of R where a stalk of the sheaf is of the form R/0P, for each prime ideal P, and . In this note we will show, this type of representation of a noncommutative ring is possible if the ring contains no nonzero nilpotent elements.

A DIVISION ALGORITHM

2005 ◽  
Vol 04 (04) ◽  
pp. 441-449 ◽  
Author(s):  
FRED RICHMAN
Keyword(s):  
Commutative Ring ◽  
Left Inverse ◽  
Division Algorithm ◽  
Nilpotent Elements

A divisibility test of Arend Heyting, for polynomials over a field in an intuitionistic setting, may be thought of as a kind of division algorithm. We show that such a division algorithm holds for divisibility by polynomials of content 1 over any commutative ring in which nilpotent elements are zero. In addition, for an arbitrary commutative ring R, we characterize those polynomials g such that the R-module endomorphism of R[X] given by multiplication by g has a left inverse.

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.

On the Uniqueness of the Coefficient Ring in a Group Ring

1983 ◽  
Vol 35 (4) ◽  
pp. 654-673 ◽  
Author(s):  
Isabelle Adjaero ◽  
Eugene Spiegel
Keyword(s):  
Commutative Ring ◽  
Cyclic Group ◽  
Group Ring ◽  
Commutative Rings ◽  
Infinite Cyclic Group ◽  
Ring Of Polynomials ◽  
Image Position ◽  
Coefficient Ring

Let R1 and R2 be commutative rings with identities, G a group and R1G and R2G the group ring of G over R1 and R2 respectively. The problem that motivates this work is to determine what relations exist between R1 and R2 if R1G and R2G are isomorphic. For example, is the coefficient ring R1 an invariant of R1G? This is not true in general as the following example shows. Let H be a group andIf R1 is a commutative ring with identity and R2 = R1H, thenbut R1 needn't be isomorphic to R2.Several authors have investigated the problem when G = <x>, the infinite cyclic group, partly because of its closeness to R[x], the ring of polynomials over R.

scholarly journals On Polynomial Extensions Of Rings

1956 ◽  
Vol 8 ◽  
pp. 1-2 ◽  
Author(s):  
Michio Yoshida
Keyword(s):  
Commutative Ring ◽  
Unit Element ◽  
Ring Of Polynomials ◽  
Polynomial Extensions

Let A be a commutative ring with unit element, and let A [x] be a ring of polynomials in an indeterminate x with coefficients in A. There are a number of well-known properties which A shares with A [x]. We shall state one of them in the following.

scholarly journals Graded near-rings

2016 ◽  
Vol 24 (1) ◽  
pp. 201-216
Author(s):  
Mariana Dumitru ◽  
Laura Năstăsescu ◽  
Bogdan Toader
Keyword(s):  
Commutative Ring ◽  
Constant Term ◽  
Graded Rings ◽  
Graded Ring ◽  
Basic Properties ◽  
Cancellative Monoid ◽  
Ring Of Polynomials ◽  
Left Cancellative Monoid

AbstractIn this paper, we consider graded near-rings over a monoid G as generalizations of graded rings over groups, and study some of their basic properties. We give some examples of graded near-rings having various interesting properties, and we define and study the Gop-graded ring associated to a G-graded abelian near-ring, where G is a left cancellative monoid and Gop is its opposite monoid. We also compute the graded ring associated to the graded near-ring of polynomials (over a commutative ring R) whose constant term is zero.

On the nil-graph of ideals of commutative Artinian rings

2018 ◽  
Vol 10 (05) ◽  
pp. 1850062
Author(s):  
T. Tamizh Chelvam ◽  
K. Selvakumar ◽  
P. Subbulakshmi
Keyword(s):  
Commutative Ring ◽  
Artinian Rings ◽  
Nilpotent Elements ◽  
Genus 2 ◽  
Vertex Set ◽  
The Ideal

Let [Formula: see text] be a commutative ring with identity and Nil[Formula: see text] be the ideal consisting of all nilpotent elements of [Formula: see text]. Let [Formula: see text] [Formula: see text] [Formula: see text]. The nil-graph of ideals of [Formula: see text] is defined as the graph [Formula: see text] whose vertex set is the set [Formula: see text] and two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text]. In this paper, we discuss some properties of nil-graph of ideals concerning connectedness, split and claw free. Also we characterize all commutative Artinian rings [Formula: see text] for which the nil-graph [Formula: see text] has genus 2.

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