# Doubt related to quotients (group or ring)

I was reading some notes about ring theory and modules and I've encountered with the following isomorphism:

$\mathbb (R[X]/ \langle x^3-1\rangle)/ \langle x-1\rangle \cong \mathbb R[X]/ \langle x-1 \rangle$ but I don't see why this is true. I've noticed that $\langle x^3-1 \rangle \subset \langle x-1 \rangle$ so I was wondering, is it true in general in the case of quotient rings that if $I \subset J$, $I,J$ ideals in $R$ then $(R/I)/J \cong R/J$?. The same doubt in the analogous case of quotient groups.

I would really appreciate if someone could explain this to me, thanks in advance.

• This is the third isomorphism theorem. – André 3000 Nov 28 '14 at 5:20
• $\dfrac{R/I}{J/I} \cong R/J.$ this is third isomorphism theorem for rings/groups. the one you mentioned are not stated correctly. $J$ is not an ideal (or normal subgroup) in $R/I.$ the image of $J$ i.e. $J/I$ is. I think there is a printing mistake or something like that in the note. – Krish Nov 28 '14 at 5:22
• How is this quotient defined why is $J$ an ideal of $R/I$? – Learnmore Nov 28 '14 at 5:29