length if and only if it has a (finite) composition series, and the length of every such composition series is equal to the length of M. Examples Finite dimensional vector spaces Any finite dimensional vector space over a field has a finite length. Given a basis there is the chainwhich is of length . It is maximal because given any chain,the dimension of each inclusion will increase by at least . Therefore, its length and dimension coincide. Artinian modules Over a base ring , Artinian modules form a class of examples of finite modules. In fact, these examples serve as the basic tools for defining the order of vanishing in intersection theory. Zero module The zero module is the only one with length 0. Simple modules Modules with length 1 are precisely the simple modules. Artinian modules over Z The length of the cyclic group (viewed as a module over the integers Z) is equal to the number of prime factors of , with multiple prime factors counted multiple times. This can be found by using the Chinese remainder theorem. Use in multiplicity theory For the need of Intersection theory, Jean-Pierre Serre introduced a general notion of the multiplicity of a point, as the length of an Artinian local ring related to this point. The first application was a complete definition of the intersection multiplicity, and, in particular, a statement of Bézout's theorem that asserts that the sum of the multiplicities of the intersection points of algebraic hypersurfaces in a -dimensional projective space is either infinite or is exactly the product of the degrees of the hypersurfaces. This definition of multiplicity is quite general, and contains as special cases most of previous notions of algebraic multiplicity. Order of vanishing of zeros and poles A special case of this general definition of a multiplicity is the order of vanishing of a non-zero algebraic function on an algebraic variety. Given an algebraic variety and a subvariety of codimension 1 the order of vanishing for a polynomial is defined aswhere is the local ring defined by the stalk of along the subvariety pages 426-227, or, equivalently, the stalk of at the generic point of page 22. If is an affine variety, and is defined the by vanishing locus , then there is the isomorphismThis idea can then be extended to rational functions on the variety where the order is defined as which is