Property of modules via exact sequences Suppose $A\neq 0$ is a commutative ring with $1$. Let $L, M, N$ be $A$-modules such that the sequence
$$0\longrightarrow L\overset{\alpha}{\longrightarrow} M\overset{\beta}{\longrightarrow} N\longrightarrow 0$$
is exact. Furthermore suppose $P$ is some property of an $A$-module (e.g. $P$ = Noetherian, Artinian, finitely generated, etc.) Now, I will say $P$ is "middle-property" if the following is true:
$$
M \textrm{ satisfies property } P \Longleftrightarrow L \textrm{ and } N \textrm{ satisfies property }P.
$$
Using this terminology, in "Undergraduate Commutative Algebra" by Miles Reid, it is proved that (in page 53) the property $P$ = Noetherian is "middle-property". I have the following questions

1) Is there standard name for what I called "middle-property"?
2) What are some other examples of "middle-property"? I have remarked
  above that being Noetherian is "middle-property". Are being Artinian, free,
  finitely-generated, flat, projective, injective, etc. are also
  "middle-property"?

I realize that I have put awful lot of questions out there. Answering any one of them is greatly appreciated :) Basically, I would like to have a list of important  "middle-properties".
Thanks.
Edit. I just realized the following: If $A$ is a Noetherian ring, then property $P$ = "finitely-generated" is also "middle property". Indeed, being Noetherian and finitely-generated are equivalent for modules over Noetherian rings (See Corollary 3.5 part (ii) in "Undergraduate Commutative Algebra" by Miles Reid, page 53). So answers that illustrate "middle-property" for particular class of rings are also welcome.

Artinian is a "middle-property". I think the rest of the properties you mention are not "middle-properties".
