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The object of this paper is to study certain class of rings called − π semi- perfect rings and generalizes this concept of modules. We call a ring R is an − π semi-perfect, if for any element Ra ∈ there is a positive integer n such that Ra n has a complement in RR , or equivalently, RaR n has a projective cover. In the first part, we have got that some of equivalent conditions to concept − π semi –perfect rings. In the second part, we generalize this concept to projective modules, we have proved that a projective module P is an − π semi-perfect if and only if, endomorphism ring of P is an − π semi-prefect. The main result of this paper the following theorem: a projective module P is an − π semi-perfect if and only if )(PJ is small in P and for any ( )RS End P φ ∈ = there is a positive integer n such that n I m φ is a direct summand of )(PJPP = and every direct decomposition of P can be lifted to a direct decomposition of P .

− π regular ring , Radical of ring , Projective cover , Projective module , Complement submodule , Regular ring

2009

مجله جامعه دمشق للعلوم الاساسيه

مجله جامعه دمشق للعلوم الاساسيه