On Skew Armendariz Rings

Structure of Zero-Divisors in Skew Power Series Rings

“…We first note that σ-rigid rings are reduced rings by [12,Proposition 3]. Using this result, we can easily check that σ-rigid rings are σ-skew power-serieswise Armendariz.…”

“…(3) σ is injective, R is reduced and σ-skew Armendariz. [12,Proposition 3] and (1) ⇒ (2) is routine. (3) ⇒ (1): Suppose aσ(a) = 0 for a ∈ R. Since R is reduced, σ(a)a = 0.…”

“…The Armendariz condition, and various derivatives described below, have been studied by numerous authors. According to Kim et al [16], we say that a ring R is power-serieswise Armendariz if a i b j = 0 for all i, j whenever power series However, we call the ring σ-skew power-serieswise Armendariz.According to Baser et al [3, Definition 3.1], a ring R with an endomorphism σ is called σ-sps Armendariz if whenever power seriesIf we use the methods of [14, Theorem 1.8], we get the fact that σ-sps Armendariz rings are σ-skew power-serieswise Armendariz, but the converse is not true (see [14, Example 1.9]).Krempa [17] called an endomorphism σ of a ring R rigid if aσ(a) = 0 implies a = 0 for a ∈ R. We call a ring R σ-rigid if there exists a rigid endomorphism σ of R. Note that any rigid endomorphism of a ring is a monomorphism and σ-rigid rings are reduced rings by [12, Proposition 3]. Using this result, we can easily check that σ-rigid rings are σ-skew power-serieswise Armendariz.…”

“…Krempa [17] called an endomorphism σ of a ring R rigid if aσ(a) = 0 implies a = 0 for a ∈ R. We call a ring R σ-rigid if there exists a rigid endomorphism σ of R. Note that any rigid endomorphism of a ring is a monomorphism and σ-rigid rings are reduced rings by [12, Proposition 3]. Using this result, we can easily check that σ-rigid rings are σ-skew power-serieswise Armendariz.…”

“…According to Kim et al [16], we say that a ring R is power-serieswise Armendariz if a i b j = 0 for all i, j whenever power series f (x) = ∞ i=0 a i x i , g(x) = ∞ j=0 b j x j in R[ [x]] satisfy f (x)g(x) = 0. In [12], a ring R with an endomorphism σ is called σ-skew Armendariz if whenever polynomials f (x) = m i=0 a i x i , g(x) = n j=0 b j x j in R[x; σ] satisfy f (x)g(x) = 0, then a i σ i (b j ) = 0 for all i, j.…”