I am trying to show the equality of two complex space germs related to double points of singular map germs $f:(\mathbb C^n,0)\to (\mathbb C^p,0)$. This two spaces are given by two (possibly non reduced) ideals, let's say $I,J$, in $\mathcal O\_{2n+s}$. I can show $J\subseteq I$ and, if we call $K$ the ideal wich defines the diagonal of $\mathbb C^n\times \mathbb C^n$ (seen into $\mathcal O\_{2n+s}$), I can show that $IK\subseteq JK$. Thus, my questions are:


(1) When does $IK\subseteq JK$ imply $I\subseteq J$?


(2) When does the equality $(JK):K=J$ hold?


(3) Would the statements in (1) and/or (2) be true, provided that $\mathcal O\_{2n+s}/K$ is regular?


Thanks!



Product and quotient of ideals
This is a partial answer. An ideal $K$ is a cancellation ideal if for any ideals $I$ and $J$, $IK=JK$ implies that $I=J$. Your question (1) says that $K$ is a cancellation ideal.
It is known that a principal ideal $K=(a)$ in a commutative ring $R$ is a cancellation ideal if and only if $a$ is not a zero divisor. 


Also by a result in the below paper $K$ is a cancellation ideal if $K$ is locally a regular principal ideal. 


D. D. Anderson and Moshe Roitman; A Characterization of Cancellation Ideals. 
Proc. Amer. Math. Soc. 125 (1997), 2853-2854 


