Let be an associative ring with identity and M be a non-zero unitary left module over . is called a hollow (semihollow) module if every proper ( finitely generated proper )submodule of is a small submodule of .
The purpose of this work is to give a comprehensive study of hollow modules and semihollow modules. Moreover we study the class of modules with finite spanning dimension.
We supply the details of the proofs for almost all the results and we illustrate the concepts by examples. Also, we add some results that seem to be new to the best of our knowledge. Among these results are the following:
- Let be a submodule of a self-projective module with , then is a small submodule of .
- Let be a module, if is a small submodule of , then has no non-zero small submodule.
- Every non-zero coclosed submodule of a hollow module is hollow.
- Let be a self-projective hollow module, then is a local ring.
- If a module has a projective cover or is a module over a max ring, then is a semihollow module if and only if is a hollow module if and only if is local.
- Let be a semihollow module with RadM is noetherian (artinian ) then is noetherian (artinian).
- Let , then M is a module with finite spanning dimension if and only if each of M1 and M2 has a finite spanning dimension.