I have edited my question according to your suggestions and comments.
Suppose now that we have a ring $R$ and a subring $S$ of $R,$ so we can define the Jacobson radicals, for example as the ideal consisting of those elements in $R$ (of $S$) that annihilate all simple right $R$-modules ($S$-modules). this radicals can be characterized in terms of maximal ideals in case that $R$ or $S$ are unital rings, and in this case $J(S)\subseteq S$.
So suppose that $R$ has unity and $S=I$ is an ideal. Then $J(I)$ Jacobson radical of $S$ is the intersection of all maximal ideals of $R$ containing $I.$ So $J(S)\supseteq S.$
This is what make me confused because with one definition $J(S)\subseteq S$ and with the other definition $J(S)\supseteq S.$ Recall also that the Jacobson radical is hereditary, that is, for any ideal $I$ of $R,$ one has that $J(I)=J(R)\cap I,$ so $J(I)\subseteq I.$
I will mantain the previous question:
Let $R$ be a ring and $I$ an ideal of $R.$ I have found that the $J(I)$ Jacobson radical of $I$ is the intersection of all maximal ideals of $R$ containing $I.$ In this case do we have $J(0)=J(R)$
This definition makes me confuse, why $J(I)$ is not the intersection of all maximal ideals of $I$ ?
Any reference will help.