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.