發(fā)布時間：2018-05-31 18:04

  本文選題：Armendariz環(huán) + 廣義詣零α-斜Armendariz環(huán)�。� 參考：《安徽師范大學》2017年碩士論文

【摘要】：Armendariz 的概念最早由 Rege 和 Chhawchharia 提出.1974年,Armendariz 證明 了:約化環(huán)是 Armendariz 環(huán),1998 年,An-derson與Camillo進一步給出了 Armendariz環(huán)的深刻結(jié)果.此后Armendariz環(huán)每年都有大量的研究成果發(fā)表,本文在此基礎(chǔ)上給出Armendariz環(huán)的三類推廣研究.首先,對π-Armendariz環(huán)的例子、性質(zhì)及相關(guān)環(huán)概念的進行深入的研究,并利用弱零化子理想補充說明了 π-Armendariz環(huán)其他性質(zhì)與相關(guān)環(huán)的關(guān)系,主要得到了: 1.若R是一個弱2-素π-Armendariz環(huán),則環(huán)R為弱zip環(huán)當且僅當環(huán)R[x]為弱zip環(huán);2.若R是一個弱2-素π -Armendariz環(huán),nil(R)是環(huán)R的一個理想,則環(huán)R為弱APP-環(huán)當且僅當環(huán)R[x]為弱APP-環(huán);3.若R是一個弱2-素π -Armendariz環(huán),nil(R)是環(huán)R的一個理想,則環(huán)R為冪零p.p.-環(huán)當且僅當環(huán)R[x]為冪零p.p.-環(huán).其次,引入廣義詣零α-斜Armendariz環(huán)的概念,對廣義詣零α-斜Armendariz環(huán)的性質(zhì)進行討論與刻畫,主要證明了: 1.設(shè)R,S是環(huán),α,β分別是R,S的自同態(tài),σ:R →S 為環(huán)的單同態(tài),且有σα = βσ.若環(huán)S是廣義詣零β-斜Armendariz環(huán),則R是廣義詣零α-斜Armendariz環(huán);2.設(shè)I是R的詣零理想,且α(I)(?)I ,則環(huán)R是廣義詣零α-斜Armendariz環(huán)當且僅當R/I是廣義詣零α-斜Armendariz環(huán).最后,引入廣義中心α-Armendariz環(huán)的概念,通過反例說明了廣義中心α -Armendariz環(huán)未必是α-弱Armendariz環(huán),并得到了如下結(jié)果:1.設(shè)α是環(huán)R的單自同態(tài),且對任意的e2 = e ∈R,α(e) = e.若 R是右廣義中心 α - Armendariz 環(huán),則 R是 abelian環(huán);2.環(huán)R是右廣義中心α: -Armendariz環(huán)當且僅當△-1R是右廣義中心α-Armendariz環(huán).
[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 偽 -skew Armendariz rings, the properties of generalized nil 偽 -skew Armendariz rings are discussed and characterized. The main results are as follows: 1. Let R _ S be a ring, 偽, 尾 be an endomorphism of R _ N _ S, 蟽 _ (1) R ~ (-1) S be a simple homomorphism of a ring, and 蟽 _ 偽 = 尾 _ (蟽). If S is a generalized nil 尾 -skew Armendariz ring, then R is a generalized nil 偽 -skew Armendariz ring. Let I be a nil ideal of R, and the ring R is a generalized nil 偽 -skew Armendariz ring if and only if R / I is a generalized nil 偽 -skew Armendariz ring. Finally, the concept of generalized central 偽 -Armendariz ring is introduced, and the counterexample shows that the generalized central 偽 -Armendariz ring is not necessarily 偽 -weak Armendariz ring, and the following result: 1 is obtained. Let 偽 be a simple endomorphism of a ring R, and for any e2 = e 鈭，

本文編號：1960791