In this note, for a ring endomorphism $\alpha$ and an $\alpha$-derivation $\delta$ of a ring $R$, the notion of weakened $(\alpha,\delta)$-skew Armendariz rings is introduced as a generalization of ...$\alpha$-rigid rings and weak Armendariz rings. It is proved that $R$ is a weakened $(\alpha,\delta)$-skew Armendariz ring if and only if $T_n(R)$ is weakened $(\bar{\alpha},\bar{\delta})$-skew Armendariz if and only if $Rx/(x^n)$ is weakened $(\bar{\alpha},\bar{\delta})$-skew Armendariz ring for any positive integer $n$.
We study the structure of the set of nilpotent elements in Armendariz rings and introduce nil-Armendariz as a generalization. We also provide some new examples by proving that if
D is a
K-algebra and
...n
⩾
2
, the coproduct
D
∗
K
K
〈
x
|
x
n
=
0
〉
is Armendariz if and only if
D is a domain with
K
∖
{
0
}
as its group of units. Finally we study the conditions under which the polynomial ring over a nil-Armendariz ring is nil-Armendariz, which is related to a question of Amitsur.
On partial-Armendariz rings 남상복; Zhelin Piao; 윤상조
Communications of the Korean Mathematical Society,
01/2019
Journal Article
Peer reviewed
This article concerns a generalization of Armendariz rings that is done by restricting the degree to one. We shall call such rings, as to satisfy this property, {\it partial-Armendariz}. We first ...show that partial-Armendariz rings are between Armendariz rings and weak Armendariz rings. The basic structures of partial-Armendariz rings are investigated, and the relations between partial-Armendariz rings and near related ring properties are also studied. KCI Citation Count: 0
For any ring R, let Nil(R) denote the set of nilpotent elements in R, and for any subset S⊆R, let Sx denote the set of polynomials with coefficients in S. Due to a celebrated example of Smoktunowicz, ...there exists a ring R such that Nil(Rx) is a proper subset of Nil(R)x. In this paper we give an example in the converse direction: there exists a ring R such that Nil(R)x is a proper subset of Nil(Rx). This is achieved by constructing a ring R with Nil(R)2=0 and a polynomial f∈Rx∖Nil(R)x satisfying f2=0. The smallest possible degree of such a polynomial is seven. The example we construct answers an open question of Antoine related to Armendariz rings.
In this article, we introduce the weak Armendariz ideals as a generalization of the Armendariz ideals and we examine its properties, its relation to other structures. Also by giving numerous examples ...and diverse, we evaluate the behavior of weak Armendariz ideals under some ring extensions../files/site1/files/71/14.pdf
The aim of this paper is to introduce and study (S, ω)-nil-reversible rings wherein we call a ring R is (S, ω)-nil-reversible if the left and right annihilators of every nilpotent element of R are ...equal. The researcher obtains various necessary or sufficient conditions for (S, ω)-nil-reversible rings are abelian, 2-primal, (S, ω)-nil-semicommutative and (S, ω)-nil-Armendariz. Also, he proved that, if R is completely (S, ω)-compatible (S, ω)-nil-reversible and J an ideal consisting of nilpotent elements of bounded index ≤ n in R, then R/J is (S, ¯ω)-nil-reversible. Moreover, other standard rings-theoretic properties are given.
In this paper we introduce and study right
Z
-Armendariz rings. A ring
R
is said to be right
Z
-Armendariz if
f
(
x
)
g
(
x
) = 0 implies that
ab
is a right singular element of
R
, where
f
(
x
) and
...g
(
x
) belong to
R
x
and
a
,
b
are arbitrary coefficients of
f
(
x
),
g
(
x
). Then we construct some examples of right
Z
-Armendariz rings by a given one. Finally, we extend this notion for modules.
On Weak Armendariz Rings Liu, Zhongkui; Zhao, Renyu
Communications in algebra,
20/8/1/, Volume:
34, Issue:
7
Journal Article
Peer reviewed
We introduce weak Armendariz rings which are a generalization of semicommutative rings and Armendariz rings, and investigate their properties. Moreover, we prove that a ring R is weak Armendariz if ...and only if for any n, the n-by-n upper triangular matrix ring T
n
(R) is weak Armendariz. If R is semicommutative, then it is proven that the polynomial ring Rx over R and the ring Rx/(x
n
), where (x
n
) is the ideal generated by x
n
and n is a positive integer, are weak Armendariz.
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