GWCN环的若干研究

发布时间：2018-07-05 07:59

  本文选题：GWCN环 + 约化环　； 参考：《安徽师范大学》2017年硕士论文

【摘要】：本文主要研究GWCN环的性质,讨论了它推广.通过研究GWCN环,一方面发现了它与一些特殊环的关系,另一方面给出了其正则性及clean性,研究了 GWCN环的扩张及推广.首先,我们介绍GWCN环和其它环的关系,并且构造若干反例说明{ CN环}(?){GWCN环} (?) {弱半交换环},得到了 GWCN环成为约化环的条件.接着讨论GWCN环的扩张,如矩阵扩张、局部化等.其次,研究了 GWCN环的正则性,证明了:(1)环R为约化环当且仅当R为CN环且R是n-正则的当且仅当R为GWCN环且R是n-正则的;(2)若R为GWCN环,则R为左弱正则环当且仅当R为弱正则环和双正则环;(3)设R是有Abelian极大左理想的GWCN环,则下列条件等价:(a) 是强正则环;(b) 为左GP-V'-环,其极大本质左理想均为广义弱理想;(c) R是左GP-V'-环,其极大本质右理想均为广义弱理想.讨论了 GWCN环的clean性,证明了:设R为GWCN环,则R为弱exchange环当且仅当R为弱clean环.最后,提出了 GWCN环的推广-α-GWCN环,讨论了它与一些特殊环的关系,研究了 α-GWCN环的一些性质,给出了:若α是环R的自同态,I为R的理想,α(I) (?) I,那么:(1)若I (?)N(R),R为α-GWCN 环,那么R/I 为α-GWCN环;(2)若I是约化的,且R/I为α-GWCN环,那么R为α-GWCN环.其中α::R/I →R/I,α(a + I) = α(a) + I,任意 a ∈ R.
[Abstract]:In this paper, we study the properties of GWCN ring and discuss its generalization. By studying GWCN rings, the relations between GWCN rings and some special rings are found, on the other hand, the regularity and clean properties of GWCN rings are given, and the extension and generalization of GWCN rings are studied. First, we introduce the relations between GWCN rings and other rings, and construct some counterexample to explain {CN rings} (?) {GWCN rings} (?) {weakly semicommutative rings}, and obtain the conditions under which GWCN rings become reduced rings. Then we discuss the extension of GWCN rings, such as matrix extension, localization and so on. Secondly, we study the regularity of GWCN rings and prove that: (1) A ring R is a reduced ring if and only if R is a CN ring and R is n- regular if and only if R is a GWCN ring and R is n- regular; (2) if R is a GWCN ring, Then R is a left weakly regular ring if and only if R is a weakly regular ring and a biregular ring. (3) Let R be a GW CN ring with a Abelian maximal left ideal, then the following conditions are equivalent: (a) is a strongly regular ring; (b) is a left GP-Va-ring. The maximal essential left ideals are all generalized weak ideals; (c) R are left GP-Va-rings, and the maximal essential right ideals are all generalized weak ideals. In this paper, we discuss the clean property of GWCN rings, and prove that if R is a GWCN ring, then R is a weak exchange ring if and only if R is a weak clean ring. Finally, the generalized 伪 -GWCN ring of GWCN ring is proposed, the relation between 伪 -GWCN ring and some special rings is discussed, some properties of 伪 -GWCN ring are studied, and the following results are given: if 伪 is the ideal of the endomorphism of ring R, 伪 (I) (?) Then: (1) if I (?) N (R) R is 伪 -GWCN ring, then R / I is 伪 -GWCN ring; (2) if I is reduced and R / I is 伪 -GWCN ring, then R is 伪 -GWCN ring. Where 伪: r / R / I / R / I, 伪 (a I) = 伪 (a) I, any a 鈭，

本文编号：2099535