A length for Artinian modules

In this paper we shall introduce a theory of length for Artinian modules over an arbitrary ring (with identity), assigning to each Artinian module A an ordinal number len(A) which will briefly be called the length of A. We also demonstrate for some familiar properties of left Artinian ring be proved efficiently using length and arithmetic properties of ordinal numbers.

Keywords:

Artinian module , Length dimension , Length function , Length rank , Strictly increasing function