/sci/ - Science & Math

I don't really understand the statement no. 5 of this ring isomorphism theorem. In particular, if [math]I[/math] is an ideal [math]I[/math] need not be a subring of R, so how do we know that [math]J[/math] is and ideal of [math]I[/math] and that it makes sense to write [math]J/I[/math]?