模M有限直和的clean性
发布时间：2019-04-01 10:41
【摘要】：设环R是有单位元的环,若环R中的元素a = e + u,其中e是环R中的幂等元,u是环R中的单位,那么称a是clean的.若环R每个元素都是clean的,那么称环R是clean环.clean环是一类重要的环,clean环的研究思想来源于模消去性问题,1977年Nicholson在研究环的exchange性时首次提出了clean环的概念.并且证明了每个clean环都是exchange环.进一步有,幂等元都是中心幂等元的环R是clean环当且仅当环R是exchange环.1994年Camillo和Yu给出了一个重要反例,说明了exchange环不一定是clean环.证明了半完全环和幺正则环都是clean的.当环不含无限正交幂等元时,半完全环,clean环和exchange环是等价的.众所周知,连续模一定是拟-连续模,拟-内射模是连续的.Mohamed和Miiller证明了由Crawley和Jonsson定义的连续模满足exchange性质.Warfield证明了模M有exchange性质当且仅当模M的自同态环是eachange环.1994年,Mohamed和Muller的结果等价于连续模的自同态环是exchange环.2006年,Nicholson等人给出了clean模的定义,并且证明了连续模是clean的.1994年,Dung和Smith给∑-CS下了定义,称模M是∑-CS的,若模M的任意直和是CS的.根据模∑-CS的定义,考虑模M的任意直和是clean的这类模,而这种模类很大,可以从模M有限直和(比如n个模M的直和)是clean的这类模入手,本文主要讨论的是模M有限直和的clean性.本文分为四个部分,第一部分是引言:第二,三部分是文章的主体部分,最后部分是结束语.第一章主要介绍了本文的研究背景和意义,给出了一些与本文密切相关的定义.如clean环和clean模的定义等.第二章是对模M有限直和的clean性的讨论.我们知道有限个clean模的直和是clean的.反之不一定,对于一些特殊的模,比如有有限不可分解的模,拟-连续模,quasi-discrete模,如果这些模的直和是clean的,那么可以证明这些模也是clean的.本章的主要结果如下:定理2.5若模M有有限不可分解的分解,那么模M是n-∑-clean的(即M(n)=M(?)...(?)M是clean的)当且仅当模M是clean的.定理2.10若模M是拟-连续模,那么模M(n)= M(?)...(?)M是clean的当且仅当模M是clean的.定理 2.18 若模M是quasi-discrete 的,那么模 M(n)= M(?)...(?)M 是clean的当且仅当模M是clean的.第三章是讨论某些满足条件(D1)的模的clean性.在探究模M有限直和的clean性的过程中,我们发现了一些满足条件(D1)的模是clean模.本章的主要结果如下:命题3.3满足条件(D1)的Rickart模是clean的.命题3.4满足条件(D1)的endoregular模是clean的.命题3.5满足条件(D1)的拟-投射模是clean的.最后部分是结束语,总结了本文的主要工作,并提出了可以进一步研究的问题.
[Abstract]:Let R be a ring with a unit, if the element a = ou in ring R, where e is the idempotent in ring R and u the unit in ring R, then a is called clean. If every element of ring R is clean, then the ring R is called clean ring. Clean ring is an important class of rings. The research idea of clean ring comes from the problem of module elimination. In 1977, Nicholson first proposed the concept of clean ring when he studied the exchange property of rings. And it is proved that every clean ring is a exchange ring. Furthermore, rings R where idempotent elements are central idempotent elements R are clean rings if and only if rings R are exchange rings. In 1994, Camillo and Yu gave an important counterexample, showing that exchange rings are not necessarily clean rings. It is proved that both semi-complete rings and unitary rings are clean's. When a ring does not contain an infinite orthogonal idempotent, the semi-complete ring, clean ring and exchange ring are equivalent. It is well known that continuous modules must be quasi-continuous modules and quasi-injective modules are continuous. Mohamed and Miiller prove that the continuous modules defined by Crawley and Jonsson satisfy exchange property. Warfield proves that module M has exchange property if and only if the endomorphism ring of module M is eachange ring. The result of Mohamed and Muller is equivalent to that the endomorphism ring of continuous module is exchange ring. In 2006, Nicholson et al gave the definition of clean module and proved that continuous module is clean. In 1994, Dung and Smith defined 鈭，
本文编号：2451492
[Abstract]:Let R be a ring with a unit, if the element a = ou in ring R, where e is the idempotent in ring R and u the unit in ring R, then a is called clean. If every element of ring R is clean, then the ring R is called clean ring. Clean ring is an important class of rings. The research idea of clean ring comes from the problem of module elimination. In 1977, Nicholson first proposed the concept of clean ring when he studied the exchange property of rings. And it is proved that every clean ring is a exchange ring. Furthermore, rings R where idempotent elements are central idempotent elements R are clean rings if and only if rings R are exchange rings. In 1994, Camillo and Yu gave an important counterexample, showing that exchange rings are not necessarily clean rings. It is proved that both semi-complete rings and unitary rings are clean's. When a ring does not contain an infinite orthogonal idempotent, the semi-complete ring, clean ring and exchange ring are equivalent. It is well known that continuous modules must be quasi-continuous modules and quasi-injective modules are continuous. Mohamed and Miiller prove that the continuous modules defined by Crawley and Jonsson satisfy exchange property. Warfield proves that module M has exchange property if and only if the endomorphism ring of module M is eachange ring. The result of Mohamed and Muller is equivalent to that the endomorphism ring of continuous module is exchange ring. In 2006, Nicholson et al gave the definition of clean module and proved that continuous module is clean. In 1994, Dung and Smith defined 鈭，
本文编号：2451492
本文链接：https://www.wllwen.com/kejilunwen/yysx/2451492.html
