A note on extensions of Baer and P. P. -rings

Armendariz, Efraim P.

doi:10.1017/s1446788700029190

Cited by 315 publications

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“…Then we also consider extending the quasi-Baer ring condition from subrings without semiprime condition, but with strengthening the intersection condition. We use our results to generalise a result of Armendariz [1]. LEMMA …”

Section: Extending the Quasi-baer Conditionmentioning

confidence: 81%

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Quasi-Baer ring extensions and biregrular rings

Birkenmeier¹,

Kim²,

Park³

2000

Bull. Austral. Math. Soc.

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A ring R with unity is called a (quasi-) Baer ring if the left annihilator of every (left ideal) nonempty subset of R is generated (as a left ideal) by an idempotent. Armendariz has shown that if R is a reduced Pi-ring whose centre is Baer, then R is Baer. We generalise his result by considering the broader question: when does the (quasi-) Baer condition extend to a ring from a subring? Also it is well known that a regular ring is Baer if and only if its lattice of principal right ideals is complete. Analogously, we prove that a biregular ring is quasi-Baer if and only if its lattice of principal ideals is complete.Throughout R will denote a ring with unity, B ( # ) its set of central idempotents, Cen (R) its centre, Z T {R) and Zi(R) its right and left singular ideals, respectively. All subrings have a unity which may not coincide with the unity of the overring. The word "ideal" used alone (that is, without the adjectives "left" or "right") means a two-sided ideal. Recall from [19] that R is a Baer ring if the right annihilator of every nonempty subset of R is generated (as a right ideal) by an idempotent. The study of Baer rings has its roots in functional analysis [3] and [19]. In [19] Kaplansky introduced Baer rings to abstract various properties of von Neumann algebras and complete regular *-rings. The class of Baer rings includes the von Neumann algebras (for example, the algebra of all bounded operators on a Hilbert space), the commutative C* -algebras C(T) of continuous complex valued functions on a Stonian space T, and the regular rings whose lattice of principal right ideals is complete (for example, regular rings which are right continuous or right self-injective). Also the Sherman-Takeda theorem [24] and [28] shows that every C* -algebra has a universal enveloping von Neumann algebra (hence a Baer ring).In [12] Clark defines a ring to be quasi-Baer if the left annihilator of every ideal is generated (as a left ideal) by an idempotent (equivalently, the left annihilator of every

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Section: Extending the Quasi-baer Conditionmentioning

confidence: 81%

“…[4] Conversely, let / be an ideal of R, and assume there exsits e 6 Si(R) such that / C eR and 1(1) n eR = eR(l -e). Let a € 1 (1). Then a = ae + a(l -e).…”

Section: Proposition 12 a Ring R Is Quasi-baer If And Only If Whenementioning

confidence: 99%

Quasi-Baer ring extensions and biregrular rings

Birkenmeier¹,

Kim²,

Park³

2000

Bull. Austral. Math. Soc.

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“…(3) Let σ be an endomorphism of a ring R with σ n (b) = b and ab = 0 for a, b ∈ R. Then we get aRb = aRσ n (b) = 0 by (2), and so R is IFP.…”

Section: Skew Ifp Ringsmentioning

confidence: 99%

Structure of Zero-Divisors in Skew Power Series Rings

Hong¹,

Kim²,

Lee³

2015

Journal of the Korean Mathematical Society

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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...

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“…R(A, B) is a left APP-ring, then A is a left APP-ring. But Example 3.9 (2) shows that B need not be a left APP-ring in general. Proof.…”

Section: Proposition 33 If a Is A Commutative Ring Then The Ring Rmentioning

confidence: 99%

“…Armendariz showed that polynomial rings over right PP-rings need not be right PP in the example in [2]. From [5, (1) R is left APP.…”

Section: Let T Be a Ring And Rmentioning

confidence: 99%

A GENERALIZATION OF PP-RINGS AND p.q.-BAER RINGS

Liu¹,

Zhao²

2006

Glasgow Math. J.

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Abstract. We introduce the concept of left APP-rings which is a generalization of left p.q.-Baer rings and right PP-rings, and investigate its properties. It is shown that the APP property is inherited by polynomial extensions and is a Morita invariant property.2000 Mathematics Subject Classification. 16D40.1. Introduction. Throughout this paper, R denotes a ring with unity. Recall that R is (quasi-) Baer if the right annihilator of every nonempty subset (every right ideal) of R is generated by an idempotent of R. In 2. Left APP-rings. An ideal I of R is said to be right s-unital if, for each a ∈ I there exists an element x ∈ I such that ax = a. Note that if I and J are right s-unital ideals, then so is I ∩ J (if a ∈ I ∩ J, then a ∈ aIJ ⊆ a(I ∩ J)). It follows from [22, Theorem 1] that I is right s-unital if and only if for any finitely many elements a 1 , a 2 , . . . , a n ∈ I there exists an element x ∈ I such that a i = a i x, i = 1, 2, . . . , n. A submodule N of a left R-module M is called a pure submodule if L ⊗ R N −→ L ⊗ R M is a monomorphism

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A note on extensions of Baer and P. P. -rings

Cited by 315 publications

References 6 publications

Quasi-Baer ring extensions and biregrular rings

Quasi-Baer ring extensions and biregrular rings

Structure of Zero-Divisors in Skew Power Series Rings

A GENERALIZATION OF PP-RINGS AND p.q.-BAER RINGS

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