Structure of Zero-Divisors in Skew Power Series Rings

Abstract

Abstract. In this note we study the structures of power-serieswise Armendariz rings and IFP rings when they are skewed by ring endomorphisms (or automorphisms). We call such rings skew power-serieswise Armendariz rings and skew IFP rings, respectively. We also investigate relationships among them and construct necessary examples in the process. The results argued in this note can be extended to the ordinary ring theoretic properties of power-serieswise Armendariz rings, IFP rings, and near-related rings.Throughout this paper every ring is an associative ring with identity. Let σ be an endomorphism of a ring R. We use R [[x; σ]] (resp. R[x; σ]) to denote the skew power series ring (resp. skew polynomial ring) with an indeterminate x over a ring R, subject to the relation xr = σ(r)x for r ∈ R. Note that σ(1) = 1 for any skew power series ring (skew polynomial ring) R [[x; σ]] (R[x; σ]), since 1x n = x n = x1x n−1 = σ(1)x n for any n ≥ 1 where 1 is the identity of R.Armendariz [2, Lemma 1] proved that whenever polynomialsover a reduced ring R satisfy f (x)g(x) = 0, then a i b j = 0 for all i, j. Rege and Chhawchharia [27] called such a ring (not necessarily reduced) Armendariz. 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.A ring R is called to satisfy the insertion-of-factors-property (simply, an IFP ring) if ab = 0 implies aRb = 0 for a, b ∈ R [4]. Narbonne In this note we will observe the previously mentioned conditions when they are equipped with an endomorphism (automorphism), and we study relationships between them. Skew power-serieswise Armendariz ringsIn this section, we study skew power-serieswise Armendariz rings which extend power-serieswise Armendariz rings and skew Armendariz rings, etc. 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. Moreover, we have the following result which gives a very short proof to Matczuk's result [21, Theorem A] and also contains a result [3, Theorem 3.3(1)]. Theorem 1.1. Let σ be an endomorphism of a ring R. Then the followi...

Insertion-of-Factors-Property Skewed by Ring Endomorphisms

“…The hypothesis "R is a σ-skew power-serieswise Armendariz ring" in Theorem 3.9(1) cannot be dropped: Indeed, the ring R which has strongly σ-skew IFP of Example 2.8 is not σ-skew power-serieswise Armendariz and R[[x; σ] Proof. We first show that R has strongly σ-skew IFP, applying the proof of [11,Lemma 3.1(2)]. Let R be a skew power-serieswise σ-Armendariz ring.…”

“…Note that every σ-skew power-serieswise Armendariz ring has σ-skew IFP by [11,Lemma 3.1(2)], but the converse is not true in general by [11,Example 3.3]. We first provide an example of a ring that has strongly σ-skew IFP and it is not σ-skew power-serieswise Armendariz as follows.…”

“…Let R be σ-skew power-serieswise Armendariz. Then R has σ-skew IFP by [11,Lemma 3.1(2)] and σ(1) = 1.…”

“…There exist many non-reduced commutative rings (e.g., Z n l for n, l ≥ 2), and many noncommutative reduced rings (e.g., direct products p(x)q(x) = 0, then a i σ i (b j ) = 0 for all i, j. It is shown that every σ-skew powerserieswise Armendariz ring has σ-skew IFP by [11,Lemma 3.1(2)].…”