Recall that a module is called 


1. semisimple if every submodule is a direct summand
2. 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?



When is every submodule pure?
Hmmm, I seem to have found the answer to this question. In


[Regular and semisimple modules](https://projecteuclid.org/journals/pacific-journal-of-mathematics/volume-65/issue-2/Regular-and-semisimple-modules/pjm/1102866791.full)


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.


