f}=(xy,x^2) \subset K[[x,y]]$, $\phi(1)=dx\wedge dy$ one has $I=(x)$, $\dim(I/J)=1$ and $B_{\phi}([x],[x])=1$.\
(iii) The above can be generalised, with almost identical proof, to the case of a one-dimensional complete intersection in an arbitrary Gorenstein ring $P$. If $\omega_P$ denotes the dualising module of $P$, then one obtains a natural pairing $$B: {\cal M} \times {\cal M} {\longrightarrow}K$$ where ${\cal M}=H_{\bf m}^0(\omega_P/J\omega_P)$.\
(iv) In the special case of hypersurfaces singularity with one-dimensional singular locus, these modules $I/J$ were studied in [@pellikaan1] under the name of [*Jacobian modules*]{} and play a rôle that can be compared to that of the Milnor ring in the isolated case. See also section 6.\
(v) In the case of hypersurface singularities with one-dimensional singular locus it is again more natural to consider the cohomology $H^n$ and $H^{n-1}$ of the $df \wedge$-complex. On the ${\mathfrak{m}}$-torsion submodule of $H^n=\Omega^n/df \wedge \Omega^{n-1}$ (isomorphic to $I/J$) there again is a pairing that does not involve any choice. In this situation there is also a map $d:H^{n-1}{\longrightarrow}H^n$ induced by exterior differentiation. That map plays a role in the Gauss-Manin system of $f$, [@vanstraten].\
(vi) Hypersurfaces in projective space with isolated singularities correspond to homogeneous singularities with one-dimensional singular locus. The module $I/J$ 