A NOTE ON CLEAN RINGS

Let M be an R-module with endomorphism ring S such that every element of S is a sum of an idempotent and a unit. If A={ƒЄ Sl ker ƒ is not contain a non-zero direct summand of MR}, then it is proved that: (i) A ﾚ B⊆ J(S). (ii)Zr (S)⊆ J(S), (iii)B=J(S) whenever MR is semi-projective, (iv) A=j(S) and the ring S/A is regular if MR is a 2-direct C2-module.

Keywords:

Clean ring , C2-module , m-direct module , regular ring