Armendariz环的三类推广研究

发布时间：2018-05-31 18:04

  本文选题：Armendariz环 + 广义诣零α-斜Armendariz环　； 参考：《安徽师范大学》2017年硕士论文

【摘要】：Armendariz 的概念最早由 Rege 和 Chhawchharia 提出.1974年,Armendariz 证明 了:约化环是 Armendariz 环,1998 年,An-derson与Camillo进一步给出了 Armendariz环的深刻结果.此后Armendariz环每年都有大量的研究成果发表,本文在此基础上给出Armendariz环的三类推广研究.首先,对π-Armendariz环的例子、性质及相关环概念的进行深入的研究,并利用弱零化子理想补充说明了 π-Armendariz环其他性质与相关环的关系,主要得到了: 1.若R是一个弱2-素π-Armendariz环,则环R为弱zip环当且仅当环R[x]为弱zip环;2.若R是一个弱2-素π -Armendariz环,nil(R)是环R的一个理想,则环R为弱APP-环当且仅当环R[x]为弱APP-环;3.若R是一个弱2-素π -Armendariz环,nil(R)是环R的一个理想,则环R为幂零p.p.-环当且仅当环R[x]为幂零p.p.-环.其次,引入广义诣零α-斜Armendariz环的概念,对广义诣零α-斜Armendariz环的性质进行讨论与刻画,主要证明了: 1.设R,S是环,α,β分别是R,S的自同态,σ:R →S 为环的单同态,且有σα = βσ.若环S是广义诣零β-斜Armendariz环,则R是广义诣零α-斜Armendariz环;2.设I是R的诣零理想,且α(I)(?)I ,则环R是广义诣零α-斜Armendariz环当且仅当R/I是广义诣零α-斜Armendariz环.最后,引入广义中心α-Armendariz环的概念,通过反例说明了广义中心α -Armendariz环未必是α-弱Armendariz环,并得到了如下结果:1.设α是环R的单自同态,且对任意的e2 = e ∈R,α(e) = e.若 R是右广义中心 α - Armendariz 环,则 R是 abelian环;2.环R是右广义中心α: -Armendariz环当且仅当△-1R是右广义中心α-Armendariz环.
[Abstract]:The concept of Armendariz was first proposed by Rege and Chhawchharia. In 1974, Armendariz proved that the reduced ring is a Armendariz ring in 1998. Anderson and Camillo further gave the profound results of Armendariz ring. Since then, a large number of research results on Armendariz rings have been published every year. On this basis, three kinds of generalized studies of Armendariz rings are given in this paper. Firstly, the examples of 蟺 -Armendariz rings, the properties of 蟺 -Armendariz rings and the concept of correlation rings are studied in depth, and the relations between other properties of 蟺 -Armendariz rings and correlated rings are explained by using weak annihilator ideals. The main results are as follows: 1. If R is a weakly 2-prime 蟺 -Armendariz ring, then R is a weak zip ring if and only if R [x] is a weak zip ring. If R is a weakly 2-prime 蟺 -Armendariz ring, then R is a weak APP- ring if and only if R [x] is a weak APP-ring 3. If R is a weakly 2-prime 蟺 -Armendariz ring, then R is a nilpotent p.-ring if and only if R [x] is a nilpotent p.-ring. Secondly, by introducing the concept of generalized nil 伪 -skew Armendariz rings, the properties of generalized nil 伪 -skew Armendariz rings are discussed and characterized. The main results are as follows: 1. Let R _ S be a ring, 伪, 尾 be an endomorphism of R _ N _ S, 蟽 _ (1) R ~ (-1) S be a simple homomorphism of a ring, and 蟽 _ 伪 = 尾 _ (蟽). If S is a generalized nil 尾 -skew Armendariz ring, then R is a generalized nil 伪 -skew Armendariz ring. Let I be a nil ideal of R, and the ring R is a generalized nil 伪 -skew Armendariz ring if and only if R / I is a generalized nil 伪 -skew Armendariz ring. Finally, the concept of generalized central 伪 -Armendariz ring is introduced, and the counterexample shows that the generalized central 伪 -Armendariz ring is not necessarily 伪 -weak Armendariz ring, and the following result: 1 is obtained. Let 伪 be a simple endomorphism of a ring R, and for any e2 = e 鈭，

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