【摘要】：設環(huán)R是有單位元的環(huán),若環(huán)R中的元素a = e + u,其中e是環(huán)R中的冪等元,u是環(huán)R中的單位,那么稱a是clean的.若環(huán)R每個元素都是clean的,那么稱環(huán)R是clean環(huán).clean環(huán)是一類重要的環(huán),clean環(huán)的研究思想來源于模消去性問題,1977年Nicholson在研究環(huán)的exchange性時首次提出了clean環(huán)的概念.并且證明了每個clean環(huán)都是exchange環(huán).進一步有,冪等元都是中心冪等元的環(huán)R是clean環(huán)當且僅當環(huán)R是exchange環(huán).1994年Camillo和Yu給出了一個重要反例,說明了exchange環(huán)不一定是clean環(huán).證明了半完全環(huán)和幺正則環(huán)都是clean的.當環(huán)不含無限正交冪等元時,半完全環(huán),clean環(huán)和exchange環(huán)是等價的.眾所周知,連續(xù)模一定是擬-連續(xù)模,擬-內(nèi)射模是連續(xù)的.Mohamed和Miiller證明了由Crawley和Jonsson定義的連續(xù)模滿足exchange性質(zhì).Warfield證明了模M有exchange性質(zhì)當且僅當模M的自同態(tài)環(huán)是eachange環(huán).1994年,Mohamed和Muller的結果等價于連續(xù)模的自同態(tài)環(huán)是exchange環(huán).2006年,Nicholson等人給出了clean模的定義,并且證明了連續(xù)模是clean的.1994年,Dung和Smith給∑-CS下了定義,稱模M是∑-CS的,若模M的任意直和是CS的.根據(jù)�！�-CS的定義,考慮模M的任意直和是clean的這類模,而這種模類很大,可以從模M有限直和(比如n個模M的直和)是clean的這類模入手,本文主要討論的是模M有限直和的clean性.本文分為四個部分,第一部分是引言:第二,三部分是文章的主體部分,最后部分是結束語.第一章主要介紹了本文的研究背景和意義,給出了一些與本文密切相關的定義.如clean環(huán)和clean模的定義等.第二章是對模M有限直和的clean性的討論.我們知道有限個clean模的直和是clean的.反之不一定,對于一些特殊的模,比如有有限不可分解的模,擬-連續(xù)模,quasi-discrete模,如果這些模的直和是clean的,那么可以證明這些模也是clean的.本章的主要結果如下: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)當且僅當模M是clean的.定理2.10若模M是擬-連續(xù)模,那么模M(n)= M(?)...(?)M是clean的當且僅當模M是clean的.定理 2.18 若模M是quasi-discrete 的,那么模 M(n)= M(?)...(?)M 是clean的當且僅當模M是clean的.第三章是討論某些滿足條件(D1)的模的clean性.在探究模M有限直和的clean性的過程中,我們發(fā)現(xiàn)了一些滿足條件(D1)的模是clean模.本章的主要結果如下:命題3.3滿足條件(D1)的Rickart模是clean的.命題3.4滿足條件(D1)的endoregular模是clean的.命題3.5滿足條件(D1)的擬-投射模是clean的.最后部分是結束語,總結了本文的主要工作,并提出了可以進一步研究的問題.
[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，

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