【作者】 刘丽；

【导师】 王尧；

【作者基本信息】 辽宁师范大学 ， 基础数学， 2011， 硕士

【摘要】 本硕士论文分为三部分:第一部分:介绍clean环的研究概述及本文的主要工作。第二部分:推广clean环的概念,提出了强f - clean环的概念,并且研究了强f - clean环上的一些性质,主要结果:性质2.2.1.3强正则环是强f - clean环。性质2.2.1.4如果R是强f - clean环,且幂等元是中心幂等元, V = R,对于任意的v∈V,存在w∈R,使v + w=0,则R通过V的平凡扩张T （ R, V）是强f - clean环。性质2.2.1.5若交换环R是强f - clean环,则R是强f - clean环。性质2.2.1.6如果R是任意交换环,则多项式环R [x ]不是强f - clean环。性质2.2.1.7如果1 = e₁ +e₂++e_n ,n≥1,其中e_i是正交幂等元,且e_i Re_i是强f - clean环,则R是强f - clean环。性质2.2.1.8设R是强f - clean环,并且e² =e∈R是中心幂等元,则e Re也是强f - clean环。第三部分:对拟clean环的研究扩张。主要结果有:性质3.5如果R是拟- clean环, W = R,对于任意的w∈R,存在t∈W,使t + w+wt+vt+wv′=0,则R通过W的理想扩张I （ R, W）是拟- clean环。性质3.6 e² =e∈R,并且e是中心幂等元,如果a∈eRe在e Re中是拟- clean的,则a在R中也是拟- clean的。更多还原

【Abstract】 We have three parts in this paper.The first part: We introduce the grand results in theclean rings and my main work in the paper.The second part: We generalize the concept ofclean rings ,and pose the concept of strongly f - cleanrings,and investigate some properties about strongly f - cleanrings.The following statement are the main results:Propertion2.2.1.3 Strongly regular rings are strongly f - clean rings.Propertion2.2.1.4 If R is strongly f - cleanrings ,and idempotents are center, V = R, -v∈V,- w∈R, make v + w=0,so the ordinary extention of T （ R, V） is strongly f - cleanrings.Propertion2.2.1.5 If Abel rings R is f - clean rings ,so R is strongly f - clean rings.Propertion2.2.1.6 If R is arbitrary Abel rings ,so polynomial rings is not strongly f - clean rings.Propertion2.2.1.7 If 1 = e₁ +e₂++e_n ,n≥1,and ei are orthogonal idempotents, and ei Re iare strongly rings, so R are strongly f - clean rings.Propertion2.2.1.8 Let R is strongly f - clean rings,and e² =e∈Ris center idempotents, e Reis also strongly f - clean rings.The third part:We investigate some properties on quasi-clean rings.The following statement are the main results:Propertion3.5 If R is quasi - clean rings, W = R, -w∈R, - t∈W,make t + w+wt+vt+wv′=0,so I （ R, V）is rings.Propertion3.6 e² =e∈R,ande is center idempotents,if a∈eReis quasi - cleanin e Re,soa is quasi - cleanin R .更多还原