# 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.

-
Aside: another useful thing to consider is "if any two have property P, then so does the third". –  Hurkyl Jun 16 '13 at 22:19
@Hurkyl: That's interesting! (Since it is a weaker condition than "middle-property", I suspect more properties would satisfy it) –  Prism Jun 16 '13 at 22:26
If $P$ is a "middle property" then the modules having property $P$ are a Serre subcategory of the category of all modules. Thick subcategory is another name used for those by some authors, as is the French name épaisse. However, other authors use the same words for the weaker condition Hurkyl is talking about. –  Martin Jun 17 '13 at 0:57
@Martin: Thanks! This essentially answers question 1). –  Prism Jun 17 '13 at 1:13

2. Artinian and noetherian modules satisfy this property, you can check this directly. The short exact sequence $0 \to \mathbb{Z} \xrightarrow{p} \mathbb{Z} \to \mathbb{Z}/p \to 0$ (for $p>1$) shows that free, projective and flat modules do not satisfy the property. It also fails for injective modules (look at $0 \to \mathbb{Z} \to \mathbb{Q} \to \mathbb{Q}/\mathbb{Z} \to 0$) and for finitely generated modules (take some non-noetherian ring $A$, some non-finitely generated ideal $I$ and look at $0 \to I \to A \to A/I \to 0$).