PURELY ? FULLY CANCELLATION MODULE

Let R be a commutative ring with identity and let M be a unital an R`-module. We introduce the concept of purely-fully cancellation R-module. An R-module M is called purely-fully cancellation if for every nonzero pure ideal I of R and every nonzero submodules N_1,N_2 of M such that IN_1=IN_2, implies N_1 =N_2. We give basic properties and characterizations about this concept. Also, we give many results to study the property of purely-fully cancellation modules in the class of multiplication modules.