Ore Extensions of Weak Zip Rings

Abstract

Abstract. In this paper we introduce the notion of weak zip rings and investigate their properties. We mainly prove that a ring R is right (left) weak zip if and only if for any n, the n-by-n upper triangular matrix ring T n (R) is right (left) weak zip. Let α be an endomorphism and δ an α-derivation of a ring R. Then R is a right (left) weak zip ring if and only if the skew polynomial ring R[x; α, δ] is a right (left) weak zip ring when R is (α, δ)-compatible and reversible.2000 MR Subject Classification. Primary 16S36, Secondary 16S99. [x; α, δ] as the Ore extension whose elements are the polynomials over R; the addition is defined as usual and the multiplication subject to the relation xa = α(a)x + δ(a) for any a ∈ R. Following Rage and Chhawchharia [14], a ring R is said to be Armendariz in that whenever polynomials f (x) = Introduction. Throughout this paper R denotes an associative ring with unity, α : R −→ R is an endomorphism and δ an α-derivation of R, that is, δ is an additive map such that δ(ab)

Weak PS-rings over Skew Hurwitz Series

“…As a generalization of the annihilator concept, Ouyang, in ( [16], 2009), introduced the weak annihilator (or nilpotent annihilator ), for a nonempty subset X of a ring R, the weak annihilator of X in R is defined as follows:…”

Special Weak Properties of Generalized Power Series Rings

“…Obviously, all reduced zip rings are weak zip, and if R is a weak zip ring, then so is the n × n upper triangular matrix ring over R. Further examples and some properties of weak zip rings are given in [15].…”

“…Any concept and notation not defined here can be founded in Ribenboim [17][18][19], Elliott and Ribenboim [6], and L. Ouyang [15][16].…”

On zip and weak zip rings of skew generalized power series

“…Motivated by the above Ouyang in [8] introduced the notion of right weak zip rings (i.e., rings provided that if the right weak annihilator of a subset X of R, Nr R (X)˝nil(R), then there exists a finite subset X 0˝X such that Nr R (X 0 )˝nil(R)), where nil(R) is the set of all nilpotent elements of R and Nr R (X) = {a 2 ROExa 2 nil(R) for each x 2 X}. The author in [8] studied the transfer of the right (left) weak zip property between the base ring R and Ore extension R[x, r, d], where r is an endomorphism and d is a r-derivation.…”

“…The author in [8] studied the transfer of the right (left) weak zip property between the base ring R and Ore extension R[x, r, d], where r is an endomorphism and d is a r-derivation. A ring R is called r-compatible if for each x, y 2 R, xy = 0 () xry = 0.…”