Nilradicals of the unique product monoid rings

2016 ◽  
Vol 16 (07) ◽  
pp. 1750133 ◽  
Author(s):  
Abdollah Alhevaz ◽  
Ebrahim Hashemi ◽  
Michał Ziembowski
Keyword(s):  
Polynomial Ring ◽  
Associative Ring ◽  
Polynomial Rings ◽  
Logical Relationship ◽  
Monoid Ring ◽  
Nilpotent Elements ◽  
Unique Product ◽  
Armendariz Rings ◽  
Coefficient Ring ◽  
Unique Product Monoid

Armendariz rings are generalization of reduced rings, and therefore, the set of nilpotent elements plays an important role in this class of rings. There are many examples of rings with nonzero nilpotent elements which are Armendariz. Observing structure of the set of all nilpotent elements in the class of Armendariz rings, Antoine introduced the notion of nil-Armendariz rings as a generalization, which are connected to the famous question of Amitsur of whether or not a polynomial ring over a nil coefficient ring is nil. Given an associative ring [Formula: see text] and a monoid [Formula: see text], we introduce and study a class of Armendariz-like rings defined by using the properties of upper and lower nilradicals of the monoid ring [Formula: see text]. The logical relationship between these and other significant classes of Armendariz-like rings are explicated with several examples. These new classes of rings provide the appropriate setting for obtaining results on radicals of the monoid rings of unique product monoids and also can be used to construct new classes of nil-Armendariz rings. We also classify, which of the standard nilpotence properties on polynomial rings pass to monoid rings. As a consequence, we extend and unify several known results.

On α-nilpotent elements and α-Armendariz rings

2015 ◽  
Vol 14 (05) ◽  
pp. 1550064
Author(s):  
Hong Kee Kim ◽  
Nam Kyun Kim ◽  
Tai Keun Kwak ◽  
Yang Lee ◽  
Hidetoshi Marubayashi
Keyword(s):  
Polynomial Ring ◽  
Polynomial Rings ◽  
Skew Polynomial Ring ◽  
Nilpotent Elements ◽  
Skew Polynomial Rings ◽  
Armendariz Rings ◽  
Skew Polynomial ◽  
Armendariz Ring

Antoine studied the structure of the set of nilpotent elements in Armendariz rings and introduced the concept of nil-Armendariz property as a generalization. Hong et al. studied Armendariz property on skew polynomial rings and introduced the notion of an α-Armendariz ring, where α is a ring monomorphism. In this paper, we investigate the structure of the set of α-nilpotent elements in α-Armendariz rings and introduce an α-nil-Armendariz ring. We examine the set of [Formula: see text]-nilpotent elements in a skew polynomial ring R[x;α], where [Formula: see text] is the monomorphism induced by the monomorphism α of an α-Armendariz ring R. We prove that every polynomial with α-nilpotent coefficients in a ring R is [Formula: see text]-nilpotent when R is of bounded index of α-nilpotency, and moreover, R is shown to be α-nil-Armendariz in this situation. We also characterize the structure of the set of α-nilpotent elements in α-nil-Armendariz rings, and investigate the relations between α-(nil-)Armendariz property and other standard ring theoretic properties.

RINGS OVER WHICH COEFFICIENTS OF NILPOTENT POLYNOMIALS ARE NILPOTENT

2011 ◽  
Vol 21 (05) ◽  
pp. 745-762 ◽  
Author(s):  
TAI KEUN KWAK ◽  
YANG LEE
Keyword(s):  
Polynomial Ring ◽  
Regular Ring ◽  
Von Neumann Regular Ring ◽  
Von Neumann ◽  
Nilpotent Elements ◽  
Von Neumann Regular ◽  
Armendariz Rings ◽  
Ring Extensions ◽  
Nil Ring

Antoine studied conditions which are connected to the question of Amitsur of whether or not a polynomial ring over a nil ring is nil, observing the structure of nilpotent elements in Armendariz rings and introducing the notion of nil-Armendariz rings. The class of nil-Armendariz rings contains Armendariz rings and NI rings. We continue the study of nil-Armendariz rings, concentrating on the structure of rings over which coefficients of nilpotent polynomials are nilpotent. In the procedure we introduce the notion of CN-rings that is a generalization of nil-Armendariz rings. We first construct a CN-ring but not nil-Armendariz. This may be a base on which Antoine's theory can be applied and elaborated. We investigate basic ring theoretic properties of CN-rings, and observe various kinds of CN-rings including ordinary ring extensions. It is shown that a ring R is CN if and only if R is nil-Armendariz if and only if R is Armendariz if and only if R is reduced when R is a von Neumann regular ring.

The McCoy condition on skew monoid rings

2017 ◽  
Vol 10 (03) ◽  
pp. 1750050 ◽  
Author(s):  
Kamal Paykan ◽  
Ahmad Moussavi
Keyword(s):  
Associative Ring ◽  
Zero Divisor ◽  
Polynomial Rings ◽  
Monoid Ring ◽  
Zero Divisor Graph ◽  
Monoid Homomorphism ◽  
Type Property

Let [Formula: see text] be an associative ring with identity, [Formula: see text] a monoid and [Formula: see text] a monoid homomorphism. When [Formula: see text] is a u.p.-monoid and [Formula: see text] is a reversible [Formula: see text]-compatible ring, then we observe that [Formula: see text] satisfies a McCoy-type property, in the context of skew monoid ring [Formula: see text]. We introduce and study the [Formula: see text]-McCoy condition on [Formula: see text], a generalization of the standard McCoy condition from polynomial rings to skew monoid rings. Several examples of reversible [Formula: see text]-compatible rings and also various examples of [Formula: see text]-McCoy rings are provided. As an application of [Formula: see text]-McCoy rings, we investigate the interplay between the ring-theoretical properties of a general skew monoid ring [Formula: see text] and the graph-theoretical properties of its zero-divisor graph [Formula: see text].

A GENERALIZATION OF NIL-ARMENDARIZ RINGS

2013 ◽  
Vol 12 (06) ◽  
pp. 1350001 ◽  
Author(s):  
MOHAMMAD HABIBI ◽  
RAOUFEH MANAVIYAT
Keyword(s):  
Sufficient Conditions ◽  
Laurent Polynomial ◽  
Polynomial Rings ◽  
Monoid Ring ◽  
Skew Polynomial Rings ◽  
Monoid Homomorphism ◽  
Special Cases ◽  
Armendariz Rings ◽  
Skew Polynomial ◽  
The Relationship

Let R be a ring, M a monoid and ω : M → End (R) a monoid homomorphism. The skew monoid ring R * M is a common generalization of polynomial rings, skew polynomial rings, (skew) Laurent polynomial rings and monoid rings. In the current work, we study the nil skew M-Armendariz condition on R, a generalization of the standard nil-Armendariz condition from polynomials to skew monoid rings. We resolve the structure of nil skew M-Armendariz rings and obtain various necessary or sufficient conditions for a ring to be nil skew M-Armendariz, unifying and generalizing a number of known nil Armendariz-like conditions in the aforementioned special cases. We consider central idempotents which are invariant under a monoid endomorphism of nil skew M-Armendariz rings and classify how the nil skew M-Armendariz rings behaves under various ring extensions. We also provide rich classes of skew monoid rings which satisfy in a condition nil (R * M) = nil (R) * M. Moreover, we study on the relationship between the zip and weak zip properties of a ring R and those of the skew monoid ring R * M.

Nilpotent Elements and Skew Polynomial Rings

2012 ◽  
Vol 19 (spec01) ◽  
pp. 821-840 ◽  
Author(s):  
A. Alhevaz ◽  
A. Moussavi ◽  
E. Hashemi
Keyword(s):  
Polynomial Ring ◽  
Sufficient Conditions ◽  
Polynomial Rings ◽  
Skew Polynomial Ring ◽  
Nilpotent Elements ◽  
Skew Polynomial Rings ◽  
Special Cases ◽  
Ring Extensions ◽  
Skew Polynomial

We study the structure of the set of nilpotent elements in extended semicommutative rings and introduce nil α-semicommutative rings as a generalization. We resolve the structure of nil α-semicommutative rings and obtain various necessary or sufficient conditions for a ring to be nil α-semicommutative, unifying and generalizing a number of known commutative-like conditions in special cases. We also classify which of the standard nilpotence properties on polynomial rings pass to skew polynomial ring. Constructing various examples, we classify how the nil α-semicommutative rings behaves under various ring extensions. Also, we consider the nil-Armendariz condition on a skew polynomial ring.

A new class of non-semiprime quasi-Armendariz rings

2014 ◽  
Vol 51 (2) ◽  
pp. 165-171
Author(s):  
Mohammad Habibi
Keyword(s):  
Polynomial Ring ◽  
Trivial Extension ◽  
Sufficient Condition ◽  
Nilpotent Ideal ◽  
Noncommutative Ring ◽  
Polynomial Extension ◽  
New Class ◽  
Monoid Ring ◽  
Armendariz Rings

Hirano [On annihilator ideals of a polynomial ring over a noncommutative ring, J. Pure Appl. Algebra, 168 (2002), 45–52] studied relations between the set of annihilators in a ring R and the set of annihilators in a polynomial extension R[x] and introduced quasi-Armendariz rings. In this paper, we give a sufficient condition for a ring R and a monoid M such that the monoid ring R[M] is quasi-Armendariz. As a consequence we show that if R is a right APP-ring, then R[x]=(xn) and hence the trivial extension T(R,R) are quasi-Armendariz. They allow the construction of rings with a non-zero nilpotent ideal of arbitrary index of nilpotency which are quasi-Armendariz.

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.

Extensions of strongly reflexive rings

2015 ◽  
Vol 08 (04) ◽  
pp. 1550078
Author(s):  
Zhaiming Peng ◽  
Qinqin Gu ◽  
Liang Zhao
Keyword(s):  
Quotient Ring ◽  
Polynomial Rings ◽  
Direct Sums ◽  
Unique Product ◽  
Reduced Ring ◽  
Unique Product Monoid

This is a further study of reflexive rings over polynomial rings and monoid rings. The concepts of strongly reflexive rings and strongly [Formula: see text]-reflexive rings are introduced and investigated. Some characterizations of various extensions of the two classes of rings are obtained. It is proved that a ring [Formula: see text] is strongly reflexive if and only if [Formula: see text] is strongly reflexive if and only if [Formula: see text] is strongly reflexive. For a right Ore ring [Formula: see text] with classical right quotient ring [Formula: see text], we show that [Formula: see text] is strongly reflexive if and only if [Formula: see text] is strongly reflexive. Moreover, we prove that if [Formula: see text] is a unique product monoid (u.p.-monoid) and [Formula: see text] is a reduced ring, then [Formula: see text] is strongly [Formula: see text]-reflexive. It is shown that finite direct sums of strongly [Formula: see text]-reflexive rings are strongly [Formula: see text]-reflexive.

Answers to Some Questions Concerning Rings with Property (A)

2017 ◽  
Vol 60 (3) ◽  
pp. 651-664 ◽  
Author(s):  
E. Hashemi ◽  
A. AS. Estaji ◽  
M. Ziembowski
Keyword(s):  
Negative Answer ◽  
Positive Answer ◽  
Left Zero ◽  
Finitely Generated ◽  
Property A ◽  
Monoid Ring ◽  
Unique Product ◽  
Zero Divisors ◽  
Unique Product Monoid

AbstractA ring R has right property (A) whenever a finitely generated two-sided ideal of R consisting entirely of left zero-divisors has a non-zero right annihilator. As the main result of this paper we give answers to two questions related to property (A), raised by Hong et al. One of the questions has a positive answer and we obtain it as a simple conclusion of the fact that if R is a right duo ring and M is a u.p.-monoid (unique product monoid), then R is right M-McCoy and the monoid ring R[M] has right property (A). The second question has a negative answer and we demonstrate this by constructing a suitable example.

NILPOTENT ELEMENTS AND NIL-ARMENDARIZ PROPERTY OF MONOID RINGS

2012 ◽  
Vol 11 (04) ◽  
pp. 1250080 ◽  
Author(s):  
M. HABIBI ◽  
A. MOUSSAVI
Keyword(s):  
Infinite Order ◽  
Monoid Ring ◽  
Nilpotent Elements ◽  
Strongly Connected ◽  
Armendariz Rings ◽  
Nil Ring

Antoine [Nilpotent elements and Armendariz rings, J. Algebra 319(8) (2008) 3128–3140] studied the structure of the set of nilpotent elements in Armendariz rings and introduced nil-Armendariz rings. For a monoid M, we introduce nil-Armendariz rings relative to M, which is a generalization of nil-Armendariz rings and we investigate their properties. This condition is strongly connected to the question of whether or not a monoid ring R[M] over a nil ring R is nil, which is related to a question of Amitsur [Algebras over infinite fields, Proc. Amer. Math. Soc.7 (1956) 35–48]. This is true for any 2-primal ring R and u.p.-monoid M. If the set of nilpotent elements of a ring R forms an ideal, then R is nil-Armendariz relative to any u.p.-monoid M. Also, for any monoid M with an element of infinite order, M-Armendariz rings are nil M-Armendariz. When R is a 2-primal ring, then R[x] and R[x, x-1] are nil-Armendariz relative to any u.p.-monoid M, and we have nil (R[M]) = nil (R)[M].

Sign in / Sign up

Export Citation Format

Share Document