(This is inspired by this question and is asked by pure curiosity).
Leave $ R $ Be a commutative ring. Leave $ M $ Y $ N $ be $ R $-modules. So, $ M $ is called $ N $-flat Yes always $ M & # 39; ? M $ it is a monomorphism of $ R $-modules, then the induced morphism. $ M & # 39; otimes_RN rightarrow M otimes_RN $ It is also a monomorphism. It can be shown that a $ R $-module $ M $ it is flat (that is, $ N $-Flat for each $ N $) yes and only if it is $ R $-flat.
Let's call a $ R $-module $ M $ flat car yes this is $ M $-flat, that is, if always $ M & # 39; ? M $ it is a monomorphism of $ R $-modules, then the induced morphism. $ M & # 39; otimes_RM rightarrow M otimes_RM $ It is also a monomorphism.
In the question mentioned above, examples were given that show that not all modules are self-supporting. On the other hand, it is easy to see that the flat modules, the injective modules and the monogenic modules are autonomous.
What are other classes of self-level modules?
(A quick search among the usual suspects (Lazard & # 39; s Autour de la platitude, From Bourbaki AC, From Lam Conferences on rings and modules, etc.) nothing about this notion appeared.)