on modukes with property (a) and theit applications

according to evans, over a commutative rings, z.d. modules have the property (a). we study modules with property (a) over noncommutative rings and prove that for a faithfull module m, the ring r⊕m has left property (a) if and only if m_r has property (a). it is shown that every indecomposable module over commutative ring has property (a). moreover, we determine several classes of modules with property (a) including modules with finite uniform dimension over commutative rings, faithful modules over semicommuative quasi-baer rings faithful flat modules over rings with left property a, faithful modules over left co-hopfian ring r with left co-hopfian ring r with left property (a), faithful modules over commutative π-regular ring, faithful modules over strongly ring.