  Let $(A,\mathfrak{m},\Bbbk)$ denote a local Noetherian ring and
$\mathfrak{q}$ an ideal such that $\ell_A(M/\mathfrak{q}M) < \infty$ for a
finitely generated $A$-module $M$. Let $\au = a_1,\ldots,a_d$ denote a system
of parameters of $M$ such that $a_i \in \mathfrak{q}^{c_i} \setminus
\mathfrak{q}^{c_i+1}$ for $i=1,\ldots,d$. It follows that $ \chi := e_0(\au;M)
- c \cdot e_0(\mathfrak{q};M) \geq 0$, where $c = c_1\cdot \ldots \cdot c_d$.
The main results of the report are a discussion when $\chi = 0$ resp. to
describe the value of $\chi$ in some particular cases. Applications concern
results on the multiplicity $e_0(\au;M)$ and applications to Bezout numbers.
