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''with respect to an ideal''

Let  $  A $
be a commutative ring with unit. A module  $  M $
over  $  A $
is said to be of finite length  $  n $
if there is a sequence of submodules (a Jordan–Hölder sequence)  $  M _ {0} \subset  \cdots \subset  M _ {n} $
such that each of the quotients  $  M _ {i} / M _ {i+ 1} $, 
$  i = 0, \dots, n - 1 $, 
is a simple  $  A $-module. (The number  $  n $
does not depend on the sequence chosen, by the [[Jordan–Hölder theorem|Jordan–Hölder theorem]].) Now let  $  M $
be an  $  A $-module of finite type and  $  \mathfrak a $
an ideal contained in the radical of  $  A $
and such that  $  M / \mathfrak a M $
is of finite length, and let  $  M \neq 0 $
be of Krull dimension  $  d $. 
(The Krull dimension of a module  $  M $
is equal to the dimension of the ring  $  A / \mathfrak q ( M) $
where  $  \mathfrak q ( M) $
is the annihilator of  $  M $, 
i.e.  $  \mathfrak q ( M) = \{ {a \in A } : {a M = 0 } \} $.) 
Then there exists a unique integer  $  e _ {A} ( \mathfrak a ;  M ) $
such that

$$ 
\textrm{length} _ {A} ( M / \mathfrak a  ^ {n+ 1} M )  = \ 
e _ {A} ( \mathfrak a ;  M ) 
\frac{n  ^ {d} }{d!}
 + \textrm{(lower degree terms)}
$$

for  $  n $
large enough. The number  $  e _ {A} ( \mathfrak a ;  M ) $
is called the multiplicity of  $  M $
with respect to  $  \mathfrak a $. 
The multiplicity of an ideal  $  \mathfrak a $
is  $  e ( \mathfrak a ) = e _ {A} ( \mathfrak a ;  A ) $. 
Thus, the multiplicity of the maximal ideal  $  \mathfrak m $
of a local ring  $  A $
of dimension  $  d $
is equal to  $  ( d - 1 ) ! $
times the leading coefficient of the Hilbert–Samuel polynomial of  $  A $, 
cf. [[Local ring|Local ring]].

There are some mild terminological discrepancies in the literature with respect to the Hilbert–Samuel polynomial. Let  $  \psi ( n) = \textrm{length} _ {A} ( M / \mathfrak a  ^ {n+ 1} M ) $
and  $  \chi ( n) = \textrm{length} _ {A} ( \mathfrak a  ^ {n} M / \mathfrak a  ^ {n+ 1} M ) $. 
Then both  $  \psi ( n) $
and  $  \chi ( n) $
are sometimes called Hilbert–Samuel functions. For both  $  \psi ( n) $
and  $  \chi ( n) $
there are polynomials in  $  n $ (of degree  $  d $
and  $  d - 1 $, 
respectively) such that  $  \psi ( n) $
and  $  \chi ( n) $
coincide with these polynomials for large  $  n $. 
Both these polynomials occur in the literature under the name Hilbert–Samuel polynomial.

For a more general set-up cf. [[#References|[a1]]].

The multiplicity of a local ring  $  A $
is the multiplicity of its maximal ideal  $  \mathfrak m $, 
$  e _ {A} ( \mathfrak m ;  A ) $.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> N. Bourbaki, "Algèbre commutative" , Masson (1983) pp. Chapt. 8, §4: Dimension {{MR|2333539}} {{MR|2284892}} {{MR|0260715}} {{MR|0194450}} {{MR|0217051}} {{MR|0171800}} {{ZBL|0579.13001}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> M. Nagata, "Local rings" , Interscience (1962) pp. Chapt. III, §23 {{MR|0155856}} {{ZBL|0123.03402}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> D. Mumford, "Algebraic geometry" , '''1. Complex projective varieties''' , Springer (1976) pp. Appendix to Chapt. 6 {{MR|0453732}} {{ZBL|0356.14002}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> O. Zariski, P. Samuel, "Commutative algebra" , '''2''' , v. Nostrand (1960) pp. Chapt. VIII, §10 {{MR|0120249}} {{ZBL|0121.27801}} </TD></TR></table>