【摘要】：作為代數(shù)學(xué)的分支,環(huán)論的重要性是不言而喻。Clean環(huán)是環(huán)論中的重要分支,從1977年W.K.Nicholson提出clean環(huán)以來,因其結(jié)構(gòu)簡單、與其它環(huán)聯(lián)系頗多的特性,逐漸被人們所重視。許多代數(shù)學(xué)者展開研究,在獲得結(jié)論的同時(shí)對(duì)其不斷的擴(kuò)張。因此,對(duì)clean環(huán)的研究是很有意義的。本文主要研究了兩類clean環(huán)的相關(guān)性質(zhì):強(qiáng)J_n-clean環(huán)與2-clean環(huán)。給出了強(qiáng)J_n-clean環(huán)性質(zhì)的等價(jià)條件、研究了單個(gè)矩陣的強(qiáng)J_n-clean性以及非交換條件下2-clean環(huán)的性質(zhì)。具體工作如下:首先,在研究強(qiáng)J_n-clean環(huán)的過程中,介紹了強(qiáng)J_n-clean環(huán)的性質(zhì)和等價(jià)條件。先是給出了強(qiáng)n-正則環(huán)來引出強(qiáng)J_n-clean環(huán)。緊接著給出強(qiáng)n-正則環(huán)和強(qiáng)J_n-clean環(huán)之間的等價(jià)刻畫。又證明了如果環(huán)R的冪等元是中心冪等的,那么R是強(qiáng)J_n-clean環(huán)當(dāng)且僅當(dāng)R/J(R)是強(qiáng)p-正則環(huán)且冪等元模J(R)可提升。還通過舉例來說明強(qiáng)J_n-clean環(huán)與其他環(huán)之間的聯(lián)系。其次,探究了單個(gè)矩陣的強(qiáng)J_n-clean性。給出了在交換條件下2′2矩陣的強(qiáng)clean性的等價(jià)條件。此外,還研究了交換局部環(huán)上2′2矩陣的強(qiáng)J_n-clean性,并利用特征方程給出2′2矩陣具有強(qiáng)J_2-clean性的判別方法。最后,在研究2-clean環(huán)時(shí),將交換條件下2-clean環(huán)的若干性質(zhì)和結(jié)論擴(kuò)展到非交換環(huán)上,給出在非交換條件下2-clean環(huán)性質(zhì)的等價(jià)刻畫。證明了非交換條件下的2-clean環(huán)與clean環(huán)之間的等價(jià)關(guān)系,并研究了非交換條件下直和、自同態(tài)環(huán)的2-clean性。此外,還考慮了矩陣環(huán)的2-clean性并進(jìn)行擴(kuò)展。
[Abstract]:As a branch of algebra, the importance of ring theory is self-evident. Clean ring is an important branch of ring theory. Since W.K.Nicholson put forward clean ring in 1977, it has been paid more and more attention because of its simple structure and many connections with other rings. Many algebraic scholars have carried out research and expanded their conclusions at the same time. Therefore, the study of clean rings is of great significance. In this paper, we study the correlation properties of two kinds of clean rings: strong J_n-clean rings and 2-clean rings. The equivalent conditions for the properties of strong J_n-clean rings are given. The strong J_n-clean property of a single matrix and the properties of 2-clean rings under non-commutative conditions are studied. The main works are as follows: firstly, in the study of strong J_n-clean rings, the properties and equivalent conditions of strong J_n-clean rings are introduced. First, strong n-regular rings are given to elicit strong J_n-clean rings. Then the equivalent characterizations between strong n-regular rings and strong J_n-clean rings are given. It is also proved that if the idempotent of a ring R is central idempotent, then R is a strong J_n-clean ring if and only if R / J (R) is a strongly pregular ring and the idempotent primitive module J (R) can be promoted. Also through, for example, the relation between a strong J_n-clean ring and other rings. Secondly, we study the strong J_n-clean property of a single matrix. In this paper, we give the equivalent conditions for the strong clean property of 2 ~ 2 matrices under commutative conditions. In addition, we study the strong J_n-clean property of 2 ~ (2) matrices over commutative local rings, and give a method to judge the strong J_2-clean property of 2 ~ (2) matrices by using the characteristic equation. Finally, in the study of 2-clean rings, some properties and conclusions of 2-clean rings under commutative conditions are extended to non-commutative rings, and the equivalent characterization of the properties of 2-clean rings under non-commutative conditions is given. The equivalent relation between 2-clean ring and clean ring under noncommutative condition is proved. The 2-clean property of direct sum endomorphism ring under noncommutative condition is studied. In addition, the 2-clean property of matrix ring is considered and extended.
【學(xué)位授予單位】：南京郵電大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2017
【分類號(hào)】：O153.3

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本文編號(hào)：2276003