ABSTRACT
Abstract Let R be a ring with identity. A non-zero unital right R-module M is compressible if M embeds in each of its non-zero submodules and is monoform if every non-zero homomorphism from a submodule N of M to M is a monomorphism. Compressible and monoform modules are investigated in the case of general rings and in the case of fully bounded rings. Relationships between the classes of compressible and monoform modules and other module classes are obtained.
26.1 Introduction Let R be a ring with identity. (In this note all rings will have an identity element and all modules will be unital right modules.) A non-zero R-module M is called compressible provided for each non-zero submodule N of M there exists a monomorphism f : M → N . (Note that Jategaonkar [17] calls a module M compressible if for each essential submodule N of M there exists a monomorphism f : M → N .) For example, if R is a (not necessarily commutative) domain then the right R-module R is compressible. In [9, Proposition 1.10], Goldie proves that if R is a semiprime right Noetherian ring then any uniform right ideal of R is a compressible right R-module. Earlier, in 1964 in fact, Goldie [8] defines, for a right Noetherian ring R and a finitely generated right R-module M , M is a basic module if M is a nonsingular compressible module and he proves that, for each right R-module M , either M is singular or M contains a basic right R-module (see [8, Theorem 3.6]).
