Fully Kasch Modules and Rings

Abstract

We carry out a study of modules MR satisfying the property that every module in σ[M] is a Kasch module. Such modules are called fully Kasch. Several sufficient conditions for a module to be fully Kasch are given which are also necessary in case the module satisfies a property (∗). We prove that if R is right Artinian, or right FBN, or Morita equivalent to a right duo ring, then every R-module satisfies the condition (∗). When R is Morita equivalent to a right duo ring, an R-module M is fully Kasch if and only if R/ ann R(mR) is a left perfect ring for any non-zero m ∈ M. These considerations tackle a question raised by Albu and Wisbauer.