If $R$ is an integral domain, then the injective hull of $R$ ($E(R)$ in symbol) is $Q(R)$. But what is $E(R)$, where $R$ is not an integral domain? In particular what is $E(R)$, where $R=k[x,y]/(x^2,xy)$, $k$ a field?
In fact, the main question for me is as follows:
Let $E=E(R)$, $Q_1=(x^2,y)/(x^2,xy)$, $Q_2=(x^2,x+y)/(x^2,xy)$. Then i guess that $(0 :_E Q_1)\neq (0 :_E Q_2)$. But i can not prove this. (perhaps it was wrong!). Can one help me? Thanks a lot.