J^(#)-U_(c)-clean环与强J^(#)-U_(c)-clean环  

J^(#)-U_(c)cleanness and strongly J^(#)-U_(c)cleanness of rings

作　　者：靳丹亚 黄涛 崔建 JIN Danya;HUANG Tao;CUI Jian(School of Mathematics and Statistics,Anhui Normal University,Wuhu 241002,China)

机构地区：[1]安徽师范大学数学与统计学院,安徽芜湖241002

出　　处：《南通大学学报（自然科学版）》2024年第1期87-94,共8页Journal of Nantong University(Natural Science Edition)　

基　　金：安徽省自然科学基金项目(2008085MA06);金融数学福建省高校重点实验室项目(JR202203);安徽省教育厅研究项目(gxyqZD2019009)。

摘　　要：设R是一个环,如果U(R)=U_(c)(R)+J^(#)(R),则称R是GU_(c)J环;如果对于任意a∈R,都存在g∈U_(c)(R),p^(2)=p∈R,d∈J^(#)(R)使得ag=p+d(且ap=pa),则称R是(强)J^(#)-U_(c)-clean环。GU_(c)J环和J^(#)-U_(c)-clean环分别是GUJ环和GJ-clean环的真推广。文章研究了GU_(c)J环的基本性质,证明了R是GU_(c)J环当且仅当R/J是U_(c)U环且U_(c)(R/J)=(U_(c)(R)+J)/J,R是U_(c)J环当且仅当R是GU_(c)J环且R/J是reduced的。此外,给出了(强)J^(#)-U_(c)-clean环的例子,得到了(强)J^(#)-U_(c)-clean环的性质和一些等价刻画,证明了若R是一个交换环,则R是GJ-clean环当且仅当存在整数n≥1使得T_(n)(R)是GJ-clean环,当且仅当存在整数n≥2使得T_(n)(R)是J^(#)-U_(c)-clean环。进一步地,研究了强J^(#)-U_(c)-clean环的Morita不变性。A ring R is GU_(c)J if U(R)=U_(c)(R)+J^(#)(R);R is(strongly)J^(#)-U_(c)-clean ring if for any a∈R,there exists g∈U_(c)(R),p^(2)=p∈R,d∈J^(#)(R),such that ag=p+d(and ap=pa).GU_(c)J rings and J^(#)-U_(c)-clean rings are proper generalizations of GUJ rings and GJ-clean rings,respectively.The properties of GU_(c)J rings are obtained.It is proved that a ring R is GU_(c)J if and only if R/J is U_(c)U and U_(c)(R/J)=(U_(c)(R)+J)/J,R is a U_(c)J ring if and only if R is GU_(c)J and R/J is reduced.Furthermore,examples,extension properties and some equivalent characterizations of(strongly)J^(#)-U_(c)-clean rings are studied,and it is proved that if R is a commutative ring,then R is GJ-clean if and only if T_(n)(R)is GJ-clean for some integer n≥1,if and only if T_(n)(R)is J^(#)-U_(c)-clean for some integer n≥2.Additionally,the study explores the Morita invariance of strong J^(#)-U_(c)-clean rings.

关 键 词：GUJ环 GJ-clean环 GU_(c)J环 J^(#)-U_(c)-clean环 强J^(#)-U_(c)-clean环 

分 类 号：O153.3[理学—数学]