强Jn-clean环和2-clean环的研究
发布时间：2018-10-17 07:49
【摘要】：作为代数学的分支,环论的重要性是不言而喻。Clean环是环论中的重要分支,从1977年W.K.Nicholson提出clean环以来,因其结构简单、与其它环联系颇多的特性,逐渐被人们所重视。许多代数学者展开研究,在获得结论的同时对其不断的扩张。因此,对clean环的研究是很有意义的。本文主要研究了两类clean环的相关性质:强J_n-clean环与2-clean环。给出了强J_n-clean环性质的等价条件、研究了单个矩阵的强J_n-clean性以及非交换条件下2-clean环的性质。具体工作如下:首先,在研究强J_n-clean环的过程中,介绍了强J_n-clean环的性质和等价条件。先是给出了强n-正则环来引出强J_n-clean环。紧接着给出强n-正则环和强J_n-clean环之间的等价刻画。又证明了如果环R的幂等元是中心幂等的,那么R是强J_n-clean环当且仅当R/J(R)是强p-正则环且幂等元模J(R)可提升。还通过举例来说明强J_n-clean环与其他环之间的联系。其次,探究了单个矩阵的强J_n-clean性。给出了在交换条件下2′2矩阵的强clean性的等价条件。此外,还研究了交换局部环上2′2矩阵的强J_n-clean性,并利用特征方程给出2′2矩阵具有强J_2-clean性的判别方法。最后,在研究2-clean环时,将交换条件下2-clean环的若干性质和结论扩展到非交换环上,给出在非交换条件下2-clean环性质的等价刻画。证明了非交换条件下的2-clean环与clean环之间的等价关系,并研究了非交换条件下直和、自同态环的2-clean性。此外,还考虑了矩阵环的2-clean性并进行扩展。
[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.
【学位授予单位】：南京邮电大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O153.3
本文编号：2276003
[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.
【学位授予单位】：南京邮电大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O153.3
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相关期刊论文 前1条
1 陈曦,刘学林,高焕英;广义逆矩阵在OFDM系统信道估计中的应用[J];无线电工程;2005年07期
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