Recall that a module is called
semisimple if every submodule is a direct summand
pure semisimple if every pure submodule is a direct summand
There is quite a bit of work on semisimple and pure semisimple modules, of course.
My question is
What is a module called if every submodule is pure?
and
What is known about these modules and where can I read about them?
Hmmm, I seem to have found the answer to this question. In
Regular and semisimple modules
by Cheatham and Smith in 1976, they call a module regular if every submodule is pure. Regular is of course an overused word, and maybe other people have called this different things. But the justification makes some sense: if I is a 2-sided ideal of R, then R/I is a regular R-module if and only if R/I is a von Neumann regular ring.
If R is a discrete valuation ring and M is an R-module such that the annihilator of M isn't zero , then every pure submodule of M is a direct summand. You can see some results about the question in the book "Modules over Discrete Valuation Rings" by Krylov and Tuganbaev.
