Example

By definition one has $e_{q,r}(M) = r^d e_q(M)$ where $d = \dim_A(M)$. Consequently, if A is local regular, one has $e_{m_A^r}(A) = r^d$. For example, if A is a discrete valuation ring, one has $e_q(A) = \operatorname{long}(A/q)$.