J#-Uc cleanness and strongly J#-Uc cleanness of rings
A ring R is GUcJ if U(R)=Uc(R)+J#(R);R is(strongly)J#-Uc-clean ring if for any a ∈R,there ex-ists g∈Uc(R),p2=p ∈ R,d ∈ J#(R),such that ag=p+d(and ap=pa).GUcJ rings and J #-Uc-clean rings are proper generalizations of GUJ rings and GJ-clean rings,respectively.The properties of GUc J rings are obtained.It is proved that a ring R is GUcJ if and only if R/J is UcU and Uc(R/J)=(Uc(R)+J)/J,R is a UcJ ring if and only if R is GUcJ and R/J is reduced.Furthermore,examples,extension properties and some equivalent characterizations of(strongly)J #-Uc-clean rings are studied,and it is proved that if R is a commutative ring,then R is GJ-clean if and only if Tn(R)is GJ-clean for some integer n≥1,if and only if Tn(R)is J#-Uc-clean for some integer n≥2.Addi-tionally,the study explores the Morita invariance of strong J#-Uc-clean rings.
GUJ ringGJ-clean ringGUcJ ringJ#-Uc-clean ringstrongly J#-Uc-clean ring
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