One sided SF-rings with certain chain conditions

Yu Fei Xiao
1994 Canadian mathematical bulletin  
We prove that with some weak chain conditions, left SF-rings are semisimple Artinian or regular. We also prove that MERT left SF-rings are really regular. A Ring R is called a left {right) SF-ring if all simple left (right) ^-modules are flat. This paper investigates left SF-rings with certain chain conditions. It shows that with some weak chain conditions, left SF-rings are semisimple Artinian rings or regular rings. The rest of this paper settles some open questions. Yue Chi Ming asked if
more » ... i Ming asked if MERT right SFrings are regular. J. Zhang and X. Du [12] answered it recently in the positive. The next question is if MELT right SF-rings are regular. J. Zhang and X. Du [12] assert that this is still open. Some recent papers show that these rings are regular if some weak conditions are added (see [9] and [12]). Here we point out that these conditions are unnecessary because MELT right SF-rings are really regular. Finally, we give an example of a left hereditary non-semisimple ring which contains an injective maximal left ideal. This settles a question proposed by Yue Chi Ming [9]. All rings throughout this paper are associative and have identities. A ring R satisfies PDCC 1 (the descending chain condition on the principal right annihilators) if there does not exist a properly descending infinite chain: r(x\ ) > r(x 2 ) > • • • > r(x n ) > • • •, for any sequence {jc n }?° C R. Similarly we may define PACC 1 , x PDCC and x PACC. A ring R satisfies left PACC (the ascending chain condition on the principal left ideals) if there does not exist a properly ascending infinite chain: Rx\ < Rx 2 < • • • < Rx n < • • •, for any sequence {x n }i° C R. Similarly we may define right PACC and left (right) PDCC. Clearly the rings satisfying left (right) PDCC are just right (left) perfect rings. When RR is /7-injective (i.e. any /^-homomorphism from a principal right ideal of R to RR can be extended to an /Miomomorphism from R R to R R ). It is easy to show that R satisfies PACC 1 (resp. PDCC 1 ) if and only ifR satisfies left PDCC (resp. right PACC). Therefore, speaking roughly, we say that PACC 1 is the dual of left PDCC etc. A ring R is called left (right) quasi-duo if all maximal left (right) ideals oïR are two-sided. R is called an MELT (resp. MERT) ring if all essential maximal left (resp. right) ideals are two-sided. R is called (Von Neumann) regular if for every x G R, there exists a y G R, such that x = xyx. J(R), Z(RR) and SOC(RR) denote, respectively, the Jacobson radical, the right singular ideal and the right socle ofR. For any subset X ofR, we define r(X) -{r G R\ Xr = 0}.
doi:10.4153/cmb-1994-040-8 fatcat:nyplhkiwnzdgleonuqdjvkb2ru