单J-内射环

Simple J-injective Rings

在文献[1]中,称环R是单J-内射环,如果对R的任意小右理想UR和任意像单的R-同态f:UR→RR,都存在c∈R,使得f=c.,但没有研究其等价刻划及扩张.论文首先给出了单J-内射环的等价条件:R是左单J-内射对任意的a∈J和R的小右理想B,r(Ra∩B)=r(a)+r(B)且任意从R的小主右理想到R的像单的同态可以定义为R中元素的右乘.其次,证明了若R是半局部,右Kasch,右单J-内射环,则:①R是左GPF环;②R是左和右Kasch环;③对任意的n≥1,Socn(RR)=Socn(RR)=l(Jn)=r(Jn);④左和右有限余生成环;⑤R是右连续环.最后,研究了单J-内射环上的几乎优越扩张.给出了若S是R的几乎优越扩张,则MS是单J-内射模MR是单J-内射环模.

In reference[1], a ring R is called a right simple J - injective if for any small right ideal UR and any R - homomorphism f：UR→RR with simple image, f = c·for some c∈R. But they haven＇t presented its equivalent conditions and extension. In this paper, firstly we give some equivalent conditions of simple J - injective ring. If R is right simple J - injective rings and only if r（Ra∩B） = r（a） ＋ r（B） for all a∈J and all small right ideals B and every R - homomorphism from a small principal right ideal of R to R with simple image can be given by right multiplication by an element of R. Secondly, we assume that if R is semilocal, right Kasch and right simple J - injective. Then ①R is left GPF. ②R is left and right Kasch. ③Socn （RR） = Soc（RR） = l（J^n） = r（J^n）for n≥1. ④R is left and right finitely cogenerated. ⑤R is right continuous. Finally, We study almost excellent extension of simple J - injective rings. We show that if S is almost excellent extension of R, then Ms is simple J - injective module =〉 MR is simple J - injective module.