Strongly Isohopfian Modules And Isomorphic Exact Sequences

The concepts of strongly isohopfian modules and rings are introduced in chapter 3.A right R-module M is said to be strongly isohopfian if,for every f∈EndR(M)and every ascending chain Kerf≤Kerf2≤1 …<Kerfn≤…,there exists an integer n≥1 such that Ker fn(?)Kerfi for every i≥n.A ring R is right strongly isohopfian if RR is a strongly isohopfian module.Basic properties of these modules and rings are studied.Let R be a semiprime ring in which amR is a complement submodule,a ∈ R,any positive integer m≥1.It is shown that R is strongly isohopfian if and only if R is strongly π-regular.The concepts of isomorphic exact sequences are posed in chapter 4,basic properties of isomorphic exact sequences are investigated.Some well known results on exact sequences are generalized.