# Ac.commutative algebra – Self-supporting modules

(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.)