Armendariz Rings and Reduced Rings

By Nam Kyun Kim, Yang Lee. 2000

266 citations recorded by the source corpus.

Throughout this paper, all rings are associative with identity. Given a w x ring R, the polynomial ring over R is denoted by R x . This paper concerns the relationships between Armendariz rings and reduced rings, w x being motivated by the results in 1, 2, 7 . The study of Armendariz rings, w x which is related to polynomial rings, was initiated by Armendariz 2 and w x Rege and Chhawchharia 7 . A ring R is called Armendariz if whenever Ž . m Ž . n polynomials f x s a q a x q ??? qa x , g x s b q b x q ??? qb x 0 1 m 0 1 n w x Ž . Ž . Ž g R x satisfy f x g x s 0, then a b s 0 for each i, j. The converse is i j . obviously true. A ring is called reduced if it has no nonzero nilpotent w x elements. Reduced rings are Armendariz by 2, Lemma 1 and subrings of Armendariz rings are also Armendariz obviously. We emphasize the connections among Armendariz rings, reduced rings, and classical quotient rings. Moreover several examples and counterexamples are included for answers to questions that occur naturally in the process of this paper.

Subfields: Algebra and Number Theory, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/60ec3111861cf4d30fb533117ae17268e7beeb64

How researchers see this paper

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In Algebra and Number Theory

1. Yang Lee
2. C. Hong
3. Yiqiang Zhou
4. Tsiu-Kwen Lee
5. Nam Kyun Kim
6. Zhong-kui Liu
7. Renyu Zhao
8. Chan Huh

In Geometry and Topology

1. C. Hong
2. Yang Lee
3. Yiqiang Zhou
4. Tsiu-Kwen Lee
5. Nam Kyun Kim
6. Zhong-kui Liu
7. Chan Huh
8. Agata Smoktunowicz

Citations in the corpus

• Armendariz and Reduced Rings (2004, methodology): “Following Krempa (1996), a rigid endomorphism of R is an endomorphism a of R such that, for any a 2 R, aaðaÞ 1⁄4 0 implies a 1⁄4 0.”
• On Skew Armendariz Rings (2003, background): “Some properties of Armendariz rings have been studied in Rege and Chhawchharia ( 1997 ), Armendariz ( 1974 ), Anderson and Camillo ( 1998 ), and Kim and Lee ( 2000 ).”
• Rings with Property (A) and their extensions (2007, background): “Following Rege and Chhawchharia [25], a ring R is called Armendariz if a i b j = 0 for all i and j whenever polynomials f (x) = (cid:5) mi = 0 a i x i and g(x) = (cid:5) nj = 0 b j x j in R [ x ] satisfy f (x)g(x) = 0 (for more detail, see [13]).”
• ARMENDARIZ RINGS AND SEMICOMMUTATIVE RINGS (2002, methodology): “In this chapter we continue the study of Armendariz rings that was done in [1, 2, 4] and [6].”
• -Clean Rings; Some Clean and Almost Clean Baer -rings and von Neumann Algebras (2010, unknown): “In general, (6) ⇒ (3) ⇒ (2) ⇒ (1), (6) ⇒ (5) ⇒ (2), and (5) ⇒ (4) ⇒ (1), and each of the implications is strict (see [10], [18] [17] and [11]).”
• A UNIFIED APPROACH TO VARIOUS GENERALIZATIONS OF ARMENDARIZ RINGS (2010, unknown): “[17] Generalizations of Armendariz rings 377”
• On Weak Armendariz Rings (2006, background): “Properties, examples and counterexamples of Armendariz rings are given in Armendariz (1974), Rege and Chhawchharia (1997), Anderson and Camillo (1998), Huh et al. (2002), Kim and Lee (2000), Lee and Wong (2003).”