# Reference Request: Finite Length Modules

Where is a good place to read about the properties of the length function on modules over a commutative ring (in particular, quotients of the ring)?

I'm looking mainly for basic properties.

I've tried looking in a number of general algebra books and commutative algebra books, but it seems that they only mention the definition along with the most basic of properties. I'd like something that goes into more depth, but so far have had no luck.

Is there a standard reference for this type of thing?

Thanks

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## 1 Answer

A nice reference is Kemper's A Course in Commutative Algebra.
You will find the basic properties of length there, but also applications to the length of $R/\mathfrak m^d$ for a local ring $(R,\mathfrak m)$ and the connection to Hilbert functions.
Actually the basics on the length function are to be found in many books on commutative algebra.

A much less known resource however is Fulton's Intersection Theory.
This is an extremely advanced, very, very difficult book but it contains a self-contained elementary algebra appendix, with many results on length which are not to be found (to my knowledge) in any other book.
You can read the appendix without fear. Don't let the body of the actual book discourage you: just ignore it!

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Thanks! I'll look into these. – user23214 Sep 7 '12 at 5:11