Armendariz rings and gaussian rings

Anderson, Derek T.; Camillo, Victor

doi:10.1080/00927879808826274

Cited by 206 publications

(148 citation statements)

References 7 publications

Supporting

Mentioning

146

Contrasting

Unclassified

Order By: Relevance

“…Then R is clearly a domain (hence Armendariz) and J(R) = pR. But R/J(R) is isomorphic to Mat 2 (Z p ) by the argument in [11,Exercise 2A] Considering Lemma 1.1 (3,5), one may naturally ask whether the converse of Lemma 1.1(5) also holds. But the answer is negative as can be seen by R = U n (A) (n ≥ 2) over a reduced ring A.…”

Section: On Radicals When Factor Rings Are Armendarizmentioning

confidence: 99%

“…A ring is called Abelian if every idempotent is central. Armendariz rings are Abelian by the proof of [3,Theorem 6] (or [20,Lemma 7]). …”

Section: On Radicals When Factor Rings Are Armendarizmentioning

confidence: 99%

“…Antoine called a ring R nil-Armendariz [4, Definition 2.3] if ab ∈ N (R) for all a ∈ C f (x) and b ∈ C g(x) whenever f (x)g(x) ∈ N (R) If a given ring R is Armendariz, then both R/N * (R) and R/N * (R) are Armendariz by Lemma 1.1 (3,5). So it is natural to ask whether R/J(R) is Armendariz if R is an Armendariz ring.…”

Section: On Radicals When Factor Rings Are Armendarizmentioning

confidence: 99%

See 2 more Smart Citations

On Jacobson and Nil Radicals Related to Polynomial Rings

Kwak¹,

Lee²,

Özcan³

2016

Journal of the Korean Mathematical Society

View full text Add to dashboard Cite

Abstract. This note is concerned with examining nilradicals and Jacobson radicals of polynomial rings when related factor rings are Armendariz. Especially we elaborate upon a well-known structural property of Armendariz rings, bringing into focus the Armendariz property of factor rings by Jacobson radicals. We show that J(R[x]) = J(R) [x] if and only if J(R) is nil when a given ring R is Armendariz, where J(A) means the Jacobson radical of a ring A. A ring will be called feckly Armendariz if the factor ring by the Jacobson radical is an Armendariz ring. It is shown that the polynomial ring over an Armendariz ring is feckly Armendariz, in spite of Armendariz rings being not feckly Armendariz in general. It is also shown that the feckly Armendariz property does not go up to polynomial rings.

show abstract

Section: On Radicals When Factor Rings Are Armendarizmentioning

confidence: 99%

“…A ring is called Abelian if every idempotent is central. Armendariz rings are Abelian by the proof of [3,Theorem 6] (or [20,Lemma 7]). …”

Section: On Radicals When Factor Rings Are Armendarizmentioning

confidence: 99%

Section: On Radicals When Factor Rings Are Armendarizmentioning

confidence: 99%

See 1 more Smart Citation

On Jacobson and Nil Radicals Related to Polynomial Rings

Kwak¹,

Lee²,

Özcan³

2016

Journal of the Korean Mathematical Society

View full text Add to dashboard Cite

show abstract

“…Now suppose that R is a left APP-ring and I = Ra 1 + · · · + Ra n is a finitely generated left ideal of R. Proof. Clearly (2) implies (1). Suppose that R is a left APP-ring.…”

Section: Definition 21 a Ring R Is Called A Left App-ring If The Lementioning

confidence: 99%

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

Liu¹,

Zhao²

2006

Glasgow Math. J.

View full text Add to dashboard Cite

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

show abstract

“…For rings that admit Armendariz or McCoy condition, it was proved in [1] (resp., [12]) that a ring R is Armendariz (resp., McCoy) if and only if R[x] is Armendariz (resp., McCoy). But it is still an open question of whether the polynomial ring over a weak Armendariz ring is weak Armendariz (see [9] …”

Section: Introductionmentioning

confidence: 99%

Extensions of linearly McCoy rings

Cui¹,

Chen²

2013

Bulletin of the Korean Mathematical Society

View full text Add to dashboard Cite

Abstract. A ring R is called linearly McCoy if whenever linear polynomials f (x), g(x) ∈ R[x]\{0} satisfy f (x)g(x) = 0, there exist nonzero elements r, s ∈ R such that f (x)r = sg(x) = 0. In this paper, extension properties of linearly McCoy rings are investigated. We prove that the polynomial ring over a linearly McCoy ring need not be linearly McCoy. It is shown that if there exists the classical right quotient ring Q of a ring R, then R is right linearly McCoy if and only if so is Q. Other basic extensions are also considered.

show abstract

Armendariz rings and gaussian rings

Cited by 206 publications

References 7 publications

On Jacobson and Nil Radicals Related to Polynomial Rings

On Jacobson and Nil Radicals Related to Polynomial Rings

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

Extensions of linearly McCoy rings

Contact Info

Product

Resources

About